We consider to migrate our DNS from DNSimple to AWS Route 53 to save some money.
DNSimple is really good in notifying you regarding expiration. Does AWS handle this also that nice?
Do I need to pay for my domains registration fees when I transfer them to Route 53?
Route 53 DNS does not expire, you pay for usage by the month.
Route 53 domain registrations, if that is what you are talking about, do expire and you need to renew each year, or better yet, just have a credit card on file and signup for auto-renewal. You are pretty much going to need to do that anyway for any other AWS services you use.
Yes, you do need to pay for your registrations when you transfer them into Route53.