It is a misquote. It is talking about the age at which the girl converted, not the age at the time of the marriage. Regarding these sorts of misquotes in general, see this related question. Regarding the significance of the age of three, it refers to a legal technicality as explained in the final bullet point of this answer. Molesting a child, whether above or below the age of three, is forbidden (Kiddushin 41a, Nidda 13b).
תניא ר' שמעון בן יוחי אומר גיורת פחותה מבת שלש שנים ויום אחד כשירה לכהונה שנאמר וכל הטף בנשים אשר לא ידעו משכב זכר החיו לכם והרי פנחס עמהם
A b'raisa: Rabbi Shim'on ben Yochai says: A female who converted to Judaism below the age of three years and one day is permissible to a kohein.1 As the verse states: "But all the female children that have not known relations with a man, keep them alive for yourselves" (B'midbar 31:18).2
This answer addresses the legal distinction between girls below and above the age of three:
The whole thing about age 3 is a technicality's technicality. With regards to certain laws, activity below the age of 3 does not affect her halachic status (for instance, a woman still has the halachic full status of "virginity" no matter what happened to her before age 3).
With regards to this case in particular: According to R' Shim'on, a Jewish girl's legal virginity status does not take effect until the age of three. Therefore, any girl who converts to Judaism below the age of three has the legal status of a Jewish virgin and may marry a kohein in R' Shim'on's opinion (see Y'rushalmi Kiddushin 4:6, "רבי שמעון אומר עד שיבואו בתולים בקדושת ישראל").
In any case, Judaism prohibits sexual relations with a minor girl, whether above or below the age of three. The Talmud states (Kiddushin 41a):
האיש מקדש את בתו כשהיא נערה: כשהיא נערה אין כשהיא קטנה לא מסייע ליה לרב דאמר רב יהודה אמר רב ואיתימא רבי אלעזר אסור לאדם שיקדש את בתו כשהיא קטנה עד שתגדל ותאמר בפלוני אני רוצה
"A man may marry off his daughter when she is a na'ara":3 When she is a na'ara, yes. When she is a child, no. This supports the teaching of Rav, for Rav Y'huda said in the name of Rav, and there are those who say Rabbi El'azar, "It is forbidden for a man to marry off his daughter when she is a child, until she grows up and says, 'I want to marry So-and-so.'"
Additionally, the Talmud specifically condemns sexual relations between men (in the context of this answer, "men" refers to all males age 13 and up) and little girls (Nidda 13b):
ת"ר הגרים והמשחקין בתינוקות מעכבין את המשיח... דנסיבי קטנות דלאו בנות אולודי נינהו
The Rabbis taught in a b'raisa: Converts4 and those who play with little girls delay the coming of the Messiah... The latter refers to those who marry [and have sexual relations with] girls who are too young to [safely]5 bear children.
The prohibition against relations between a man and a minor girl not of safe childbearing age is codified in the Rambam (Mishneh Torah, Issurei Bi'ah 21:18) and the Shulchan Aruch (EH 23:1).6
As far as conducting a formal marriage (kiddushin, though without consummation of the marriage) between a man and a minor girl, the Rambam (Ishus 3:19) and perhaps the Shulchan Aruch (EH 37:8; see also Kesef Mishneh, Ishus 10:16) rule that the Rabbis merely discouraged the practice and that it is not technically prohibited. Nonetheless, this is only technically permitted if the girl is at least close to the age of majority, both the girl and her father approve of the marriage, and if the girl is mentally mature and competent to make such a decision (see Bach EH 37 and Chelkas M'chokeik EH 37:10).
However, Rashi (K'subos 57b, s.v. "אבל פוסקין") understands that marriage to a minor girl is formally prohibited, though betrothal with the consent of both the girl and her father is permissible. The Rivash also rules that marriage to a minor girl is formally prohibited (§193, "דדרך בני אדם להקפיד שלא להשיא בנותיהן בקטנותן וגם שיש לו אסור בדבר מדברי רב יהודה עד שתגדיל"; see also §199), as does the Taz (EH 37:5, based on the Ran).
Regardless of the precise nature of the talmudic exhortation not to marry minor girls, some communities during medieval times (and even as late as the 17th century)7 felt the need to marry off their daughters as soon as they could, even while they were children (see Tosafos on Kiddushin 41a, s.v. "אסור לאדם שיקדש את בתו"; Rama EH 37:8). The reason for this was the extremely precarious nature of the safety and finances of Jews in Europe during that period; if a father acquired enough capital to pay his daughter's dowry, he would immediately marry her off before her marriage prospects could be compromised by his money being stolen or confiscated by the state - or worse.
In any case, later authorities have decried the practice of marrying off one's minor daughter, and they have stressed the prohibition against it. R' Ya'akov Emden (18th century, Sh'eilas Ya'beitz I, §14, "לדידן לא שנא קידושין וכ"ש נישואין בתרווייהו לא שפיר דמי למיעבד עובדא בקטנה ממש") and the Aruch HaShulchan (19th c, EH 37:33, "בזמנינו אין מקדשין את הקטנות ולא שמענו מעולם מי שיקדש את הקטנה ונכון הוא דבש"ס [מ"א א] איתא דאסור לקדשה כשהיא קטנה... אין זה נכון וכן המנהג פשוט בזמנינו במדינתינו") have ruled that it is wrong for a man to marry a minor in this day and age regardless of leniencies that were employed in some communities during the Middle Ages.
In the social and psychological context of modern times, it is known that it would be psychologically damaging and traumatic for a minor girl (or even a girl well past the age of 12) to be married. Leaving aside the technical considerations discussed heretofore in this answer, putting a young girl in a marriage or sexual relationship would therefore be a violation of multiple tenets of Judaism, including the precept to love one's fellow as oneself (Vayikra 19:18), to do that which is just and good (D'varim 6:18), and not to torment one's fellow (Vayikra 25:17).
Additionally, although there are various opinions as to the extent of this obligation (see this answer and answers to this question), Jewish law requires a Jew to observe secular law in the jurisdiction to which the Jew is subject. This would be yet another reason to prohibit such relationships in this day and age.
Circling back around to your question about that website misrepresenting the Talmud and Jewish law, the website was wrong in multiple ways.
The halacha in this case does not follow R' Shim'on but instead follows the Rabbis (see Rambam, Hil. Issurei Bi'ah 18:1).
A child under three cannot be married to a kohein or anyone else. The age of three has technical relevance as discussed in section I. above.
According to R' Shim'on, woman who had converted to Judaism before the age of three can marry either a kohein or a non-kohein, and there is no reason for a kohein to prefer her over anyone else. (According to the Rabbis, only a non-kohein may marry her).
Talmudic Jews determine the halacha not simply by reading Tannaitic literature, but by considering the entire corpus of Jewish law all the way until contemporary times.
Jewish law does not endorse pedophilia, and the Talmud castigates those who would "play" with young girls. Although teenage marriage was common during much of Jewish history, and some medieval communities took the legally controversial approach of marrying off their preteen daughters, Jewish law would not sanction behavior that is understood to be psychologically unhealthy, traumatic, or illegal.
The website that produced the claims in your question is anti-Semitic and anti-Israel. This answer by Danno discusses this general sort of attack on the Talmud and Judaism:
These attacks are usually amalgamations of the following:
Pure invention -- some of the books listed don't exist or the quotes are fabrications
Mistranslations or selective quoting
Out of context quotes (statements made in the course of a protracted legal argument presented as definitive statements of belief or statements made to make a legal point being cited as normative practice).
There are plenty of sites which go through "quotes" like these and explain them one at a time. People who cite the attack pages rarely learn or read the original material or read the responses and understand the legal subtleties involved.
1 Literally, the translation is, "A female convert below three years and one day is permissible for the k'huna." That the meaning is as rendered in the body of this answer is evident from the context of the subsequent talmudic passage as well as the classic Talmudic commentators (e.g. Rashi, s.v. "כשרה לכהונה"). Additionally, the Jerusalem Talmud (Kiddushin 46a) explicitly renders the b'raisa's quote as, "a female convert who had converted below three years and one day..." (" תני בשם ר' שמעון גיורת שנתגיירה פחותה מבת שלש שנים ויום אחד כשירה לכהונה").
The Talmud brings R' Shim'on's opinion to contrast with the opinion of the other Rabbis who rule that a kohein may not marry a convert regardless of the age at which she converted; according to the Rabbis, all female converts have the status of zona with respect to marrying a kohein.
2 This verse describes the aftermath of the Israelites' retributive war against Midian (B'midbar 31). Moshe commanded the Jewish people to slay all Midianite males and all females of age three and above, as Midian participated in Bil'am's scheme to seduce the Jewish men and entice them to worship idolatry. According to R' Shim'on bar Yochai's opinion, the phrase "keep them alive for yourselves" implies that all female Midianite captives who were captured below the age of three could eventually be suitable marriage partners for all the soldiers in the war. Since Pinchas was a kohein, and he participated in the war, this implies that these women would be permissible for marriage to kohanim. The Talmud continues and states that R' Shim'on's colleagues interpreted "keep them alive for yourselves" differently, namely that the captives could be taken as maidservants.
3 Lit. 'maiden.' This refers to a legal category when a female is both at least 12 years old and within the first six months of having shown signs of puberty (Kiddushin 16a; see also, for example, Chidushei Ramban on Y'vamos 61b, "י"ל מדכתיב והנערה ש"מ לא בגרה עדיין אלא שהתה מלהביא שתי שערות שהן סימני נערות עד י"ד").
4 Although the context of the Talmudic passage makes it clear that it is condemning men who marry children, there are different ways to interpret why the Talmud says that converts delay the coming of the Messiah. See this chat comment, as well as this and subsequent chat comments for various interpretations mentioned in Tosafos on Kiddushin 70b, s.v. "קשים גרים".
5 See Y'vamos 12b.
6 The Ritva (13th c, Y'vamos 61b, "
וסוגיין הכא ובכל דוכתין דמותר לבעול אשה שאינה בת בנים כלל... וכדשרו בכל דוכתי נשואי קטנה וזקנה"), Tosafos (Rabbeinu Tam, 12th c, Y'vamos 12b, s.v. "שלש נשים"), the Mordechai (13th c, ad loc.), the Rama (16th c, EH 31:5), and possibly the Rosh (13th c, Responsum 33:3), however, disagree with the prohibiting opinions of the Shulchan Aruch, Rambam, Rashi, Rivash, and Ran. These former opinions (Ritva, et al.) maintain that sexual relations with a minor girl are not inherently prohibited in the context of marriage, and that both the marriage and the relations would merely be discouraged (per the talmudic sources mentioned above). However, see section III. in the body of this answer for the practical halacha.
7 See, for example, the ruling of R' Chaim Benveniste in K'nesset HaG'dola (EH 37:9).