The simple answer to your question, Naomi, is that nobody yet knows why autism rates have increased. However, there is even debate about whether autism rates have increased at all, or whether it's merely a relic of increased awareness and/or changes in diagnostic practice.
https://www.annualreviews.org/doi/10.1146/annurev.publhealth.28.021406.144007
What *is* understood, though, is that there has been no observed link between vaccinations and autism. Once again, I can only refer you back to the study in Japan, which has had the unique opportunity to review autism rates before and after the withdrawal of MMR. As you will have seen, there was no decrease after the withdrawal, and in fact rates continued to rise. Coupled with evidence gathered across the world from multiple studies and bodies, it should be clear by now that whatever the causes of autism, it has nothing to do with vaccination.