This halacha is brought in Shulchan Aruch 53:16
Here are some clarifications:
1) A regular/permanent Shliach Tzibur does not need to refuse.
(On the contrary, if a regular Shliach Tzibur refuses he shows that it is beneath his dignity
to daven until the congregation pleads with him [see
Beit Yosef and
Shulchan Aruch Harav 53:19])
2) If requested by an 'Adam Gadol' (say, the Rav of the shule ) - one does not need to refuse.
3) The Eishel Avraham (Butchatch) writes that this law only applies when you really feel unworthy of the great honor of being chosen to lead the congregation, e.g if there are other candidates of Shliach Tzibur worthier than you, and only after being convinced by the Gabai (say) that you are also worthy - only then can you agree.
4) Nowadays (especially for weekday tefilot) , the role of the Shliach Tzibur is not associated with honor - so (except for special tefilot - like Rosh Hashana etc) there is no reason to refuse.
On the contrary if you see that no one is interested to be Shliach Tzibur it's a mitzvah to appoint yourself - as he is preventing a Tircha DeTzibura. (Tshuvot VHanhagot Chayai Adam [Siman 112])
5) Similarly, if you think that someone not fit is going to be Shliach Tzibur - then you should appoint yourself. (Eishel Avraham [see above])
(This answer is mostly a summary from the sefer Piskei Tshuvot p464)
PS: In short, if you think that you are pretty much worthy to be the Shlich Tzibur (for a regular davening) and there's no permanent Shliach Tzibur - go for it! (at least that's my understanding)
This excerpt of Tanya is also relevant:
ועיקר הסיבה וגרמא בנזקין הוא מהיורדים לפני התיבה שהוא הפקר לכל הרוצה
לפשוט רגליו החוטף אפרתי או מחמת שאין גם אחד רוצה וכו'. ואי לזאת זאת
העצה היעוצה ותקנה קבועה חוק ולא יעבור עוד ח"ו דהיינו לבחור אנשים
קבועים הראוים לזה עפ"י הגורל או בריצוי רוב המנין.