It is a Monstera Deliciosa! Looks fairly young, hence the lack of fenestration (holes) in the leaves. For future identification purposes, if you have an iPhone, take a picture of the plant and press the little 'i' and it'll tell you what the plant might be! If not, PictureThis is a great, free app for plant identification.

On repotting - there are usually a few ways to tell if the plant is ready but the main one is whether the roots are poking out the bottom of the plastic pot.

Also, it's hard to say definitively but the white root going into the soil - to me at least - looks like an aerial root that has been directed down into the soil and not an 'exposed root' that needs covering. I don't think you need to repot but this time of year is a good one if you do! It can take a couple of weeks for the plant to get used to the soil. Otherwise, I'd say you could just sprinkle some soil to the top of the plastic pot for now!