In May, my professor agreed to write me letter of recommendation, but now I just graduated I just graduated recently in May, and right before graduation, I asked if my professors could write me LOR for grad school. And they all agreed. Now I'm in the process of applying to grad schools, but it's almost October, and how should I write the email to remind my professors of LOR? Should I start by asking if their offers still stand, and if they need anything else (like supporting materials) from me?
Be friendly, normal, polite, and human and remember that your professors are (or at least should be) too.
If they offered to write the LORs, they want to help you out. It makes people feel good to help others especially with LORs, because you get to help someone and feel nice and authoritative.
I wouldn't even bother asking if the offer still stands. Just make sure that you tell them that you're grateful and appreciative that they made the offer, let them know if there are any deadlines that you have coming up which make the need for LORs more urgent, and you should be just fine. 
Always offer supplemental documentation if they need it.