Why do Sefardim start saying Sleichos earlier than Ashkinazim?
The simple answer is because Shulhan Arukh 581:1 and the Arizal Sha'ar HaKavvanot 89d, Pri Etz Haim 128b(see also Likutei Torah 106b, Shaar Ruah HaKodesh 48a and Shaar HPesukim 41a) say so. For a more thorough answer according to halakha and Kabbalah see Divrei Shalom with perush Ner Shalom by R' Shalom Afjin(unfortunately not online). He essentially details the custom back to the time of the Geonim. It is a good amount of Pilpul as there were Geonim both for and against and the same with the Rishonim.