Why am I obligated to do Teshuvah if I do a חטא (sin) accidentally, according to Rambam (Hilchot Teshuva 1:1)? What am I doing Teshuvah for?
The soul is incapable of doing wrong, even by mistake. Such שגגות are possible because of previous sins committed knowingly and by choice.
Based on Rav Ovadya Yosef at http://www.halachayomit.co.il/displayRead.asp?ReadID=1096
The Ramban (in Toras Ha'adam Shaar Hegemul) brings two reasons why one who sins בשוגג requires repentance:
1) If he was truly worried and fearful of Hashem he would not have done what he did without investigating fully if it was permitted. I remember hearing an example for this that you don't hear of anyone accidentally jumping into a fire. This is also explained in the Rambam (Hilchos Shegogos Chapter 5 Halacha 6) who differentiates between a שוגג and an אונס. In the case of a שוגג the person is at fault for not bothering to thoroughly investigate, whereas in an אונס situation there was nothing he could have done.
2) Even if the person cannot be blamed for his action, the end result is that the sin was committed which affected a blemish on his soul that must be rectified. This is brought out in more "lomdishe" words by the Toras Chesed (O.C. Siman 31) who explains that there are two parts to commiting a sin: the גברא (person) and the חפצא (object). The גברא part means that the individual is commanded to fulfill the will of Hashem, and by failing to do so he is rebelling, and the חפצא part is that by performing an aveirah a blemish is caused to his soul. In the case of a שוגג, although the גברא issue is not present (he is not rebelling since he did not intend to do so), the חפצא part is still present and needs atonement.
[Based on this reasoning some Achronim (Nesivos Hamishpat C.M. Siman 234 Biurim Sif Kotan 3, and see Ohr Sameach Hilchos Girushin Perek 1 Halacha 17) claim that by a Rabinical prohibition which is only on the גברא not the חפצא one does indeed not require attonment for a שוגג.]
(Answer based on this Hebrew essay.)