Concerning your question on why these are the three cardinal sins, the Gemara learns us the following (Arakhin 15b):
With regard to forbidden sexual relations it is written that when Potiphar’s wife attempted to seduce Joseph he responded: “How can I do this great wickedness, and sin against G-d” (Genesis 39:9). With regard to bloodshed it is written, after Cain murdered his brother: “And Cain said to the L-rd: My punishment is greater than I can bear” (Genesis 4:13). The Torah describes each of these three cardinal sins with the word “great” in the singular, whereas malicious speech is described with the plural term “great things,” indicating that it is equivalent to all three of the other transgressions together.
So, the Gemara is teaching us that the Torah uses the word "great" in describing these sins.I'll demonstrate it in the posuks below:
Incest:With regard to forbidden sexual relations it is written that when Potiphar’s wife attempted to seduce Joseph he responded: “How can I do this great wickedness, and sin against G-d” (Genesis 39:9)
Murder: With regard to bloodshed it is written, after Cain murdered his brother: “And Cain said to the L-rd: My punishment is greater than I can bear” (Genesis 4:13).
Idolatry: And it is written with regard to idol worship: “And Moses returned to the Lord, and said: Oh, this people have sinned a great sin, and have made for themselves a god of gold” (Exodus 32:31).
Concerning lashon hara, the Gemara explains further:
The Gemara asks: Granted that with regard to malicious speech the verse uses the plural: “Great things,” but the plural indicates a minimum of two. If so, one can only say that malicious speech is equivalent to two of the cardinal transgressions. The Gemara responds: Which of them could be taken out as less than the other two? All three are equal. Therefore malicious speech must be equivalent to all three.
The base for lashon hara in this Gemara is found in a posuk in Tehillim:
“May the L-rd cut off all flattering lips, the tongue that speaks great things” (Psalms 12:4).
Extra sources: תלמוד בבלי, see also footnote 31-32 in sefer הקדמת הרוקח:
אולי עפ"י מדרש תהלים מזמור יב יכרת ה' כל שפתי חלקות לשון מדברת גדולות תהילים יב ד בר קפרא בשם רבי יונתן אמר קשה לשון הרע שכבר בא דוד וכרתו ברוח הקודש שנאמר יכרת ה' כל שפתי חלקות קשה לשון הרע משלש עבירות שנקראו גדולות עבודה זרה וגלוי עריות ושפיכות דמים בעבודה זרה כתיב אנא חטא העם הזה חטאה גדולה שמות לב לא בגילוי עריות כתיב ואיך אעשה הרעה הגדולה הזאת בראשית לט ט בשפיכות דמים כתיב גדול עוני מנשוא בראשית ד יג ובלשון הרע כתיב יכרת ה' כל שפתי חלקות לשון מדברת גדולות"' אולי עפ"י אבות ג יג רבי עקיבא אומר שחוק וקלות ראש מרגילין
See also: Shaarei Teshuva 3:201-209
The Gemara elsewhere (Sanhedrin 74a) explains:
With regard to all other transgressions in the Torah, if a person is told: Transgress this prohibition and you will not be killed, he may transgress that prohibition and not be killed, because the preserving of his own life overrides all of the Torah’s prohibitions. This is the halakha concerning all prohibitions except for those of idol worship, forbidden sexual relations, and bloodshed. Concerning those prohibitions, one must allow himself to be killed rather than transgress them.
The reason why there are three cardinal sins, and these are incest, murder and idolatry, I would like to begin explaining, by bringing a Rambam.
In his Mishneh Torah; Hilchos Yesodei HaTorah 5:5 he explains:
Similarly, if gentiles told [a group of Jews]: "Give us one of you to kill. If not, we will kill all of you," they should allow themselves all to be killed rather than give over a single soul to [the gentiles].
Concerning forbidden sexual relationships, the Torah compares this to actually murdering someone:
But if a man find a betrothed girl in the field, and the man force her, and lie with her: then only the man that lay with her shall die: See Yoma 86a
Concerning idolatry, during Shema, we read "And thou shalt love the L-rd thy G-d with all thy heart, and with all thy soul, and with all thy might." (Devarim 6:5).
On this posuk in Devarim 6:5, the Gemara in Yoma 86a explains that "with all thy soul" (as we read in the Shema), actually means that:
Rather, it is to teach that if there is a person whose body is more beloved to him than his property, therefore it is stated: “With all your soul.” The verse teaches that one must be willing to sacrifice his life to sanctify God’s name.