If masturbation is, as stated in the Shulchan Oruch (Even HaEzer 23:1), "the worst sin in the Torah", then why is it not listed in any of the various lists of 613 Mitzvos and why is there no verse in the Torah that mentions it directly ("Do not masturbate")?
Chazal would emphasize through exaggeration (דברו חכמים לשון הבאי). (Tamid 29a) One of the examples given there is that the Mishna says that they let the animal for the tamid sacrifice drink from a gold cup (so that it would be hydrated and the carcass would be easier to manipulate). In reality, Rava argues, a copper cup was used and the Mishna was merely expressing the service in terms of magnificence. Another example of Chazal using לשון הבאי can be found in the Teshuvos HaGeonim (Musafia, 26).
Chazal would frequently employ a similar device when denouncing certain bad behavior, for example:
ארבעה דברים שהן נפרעין מן האדם בעולם הזה והקרן קיימת לו לעולם הבא ואלו הן ע"ז גילוי עריות ש"ד ולשון הרע כנגד כולן (Yeushalmi, Peah 4a)
אמר רבי יוחנן משום ר' שמעון בן יוחי נוח לו לאדם שיפיל עצמו לתוך כבשן האש ואל ילבין פני חבירו ברבים (Sotah 10b)
וכבר הזהירו חכמים מאד על ההרהור והרחיקו מגורמיו, והאריכו לירא ולהפחיד מקשה עצמו לדעת, ומוציא שכבת זרע לבטלה, וביארו שכל זה אסור, אבל לא חייבו מלקות בשום דבר מסוג זה.
Likewise, the Beis Shmuel (Even HaEzer, 23:1) is of the opinion that statement of the Shulchan Aruch cited above in the original post is not meant literally.
Some opinions hold that this is a Biblical prohibition, though there is disagreement as to the primary Biblical source of the prohibition (possible candidates include bal tashchis, v'nishmarta mikol davar ra, and others). (For example: Based on the severity of Talmudic and rabbinic statements on this topic, as well as on the Biblical source discussed by Tosafos [Avoda Zara 20b], Rabbi Moshe Feinstein holds that it must be an outright Biblical prohibition). The Tzitz Eliezer cites other opinions that variously maintain that the prohibition either is or may be Rabbinic with Biblical allusions (asmachta b'alma).
Your first question, "why is it not one of the 613 mitzvos", assumes that all important commandments are part of the 613. This is not the case. The Behag, for example, does not list belief in God as one of the 613, and Ramban explains because this is a foundational commandment, therefore it is not listed. Similarly, the obligation to develop one's character is a foundational commandment, and therefore is not included (per R. Chaim Vital). There are other commandments which are not listed in the 613 for different reasons.
Additionally, according to Rabbi Jacob Tam, this prohibition is included in the mitzva of procreation, thus it is part of the 613. And Smag counts it as an explicit commandment in his list, as other responders have noted.
Your second question, "why is there not an explicit verse", again is based on the premise that all important commandments are explicitly stated. Again, this is not so. There are indisputably important commandments which are not explicit, for example the annulment of vows, the sacrifice offering of a convert, the prohibition of bypassing one mitzva for another, seclusion with a forbidden woman, and many others.
I will stick to the Kra. I think it mentioned implicitly in the Torah:
Last Shabbat we read in the beginning of Parashat Kedoshim:
דבר אל כל עדת בני ישראל ואמרת אלהם - קדשים תהיו כי קדוש אני ד' אלקיכם
("Speak unto all the congregation of the children of Israel, and say unto them: Ye shall be holy; for I HaShem your G-d am holy").
Rashi Mefaresh this as:
הוו פרושים מן העריות ומן העבירה, שכל מקום שאתה מוצא גדר ערוה אתה מוצא קדושה
That is to one can reach Kedusha by refraining himself from the Arayot. Clearly, if a man MeKadesh himself by refraining from excessive intercourse habits he should Kadesh himself by refraining from masturbation.
That being said, we are shown how the Torah relates to masturbation in general: BeReshit 38, 9:
וידע אונן כי לא לו יהיה הזרע והיה אם בא אל אשת אחיו ושחת ארצה לבלתי נתן זרע לאחיו. וירע בעיני ד' אשר עשה וימת גם אותו
("And Onan knew that the seed would not be his; and it came to pass when he went in unto his brother's wife, that he spilled it on the ground, lest he should give seed to his brother. And the thing which he did was evil in the sight of HaShem; and He slew him also.").