Yikes, use your imagination....they'll get into the insulation, chew on wires, defecate all over, breed, etc. I assure you that if they can get in, they can also get out. Trapped inside without food and water, they may make new exit holes.

Get those traps out ASAP.

Have you seen the rats?
I ask because we suffered with Red Squirrels, they set up nest sites and breed youngsters.
In the end I trapped 6 of the pests. Live traps are the only answer, baited with Peanut butter.
You''ll need to seal the exit once you get them out. I sealed it before I trapped them, all they did was to make another hole.
I'll be surprised if they're rats, as they normally live under ground. There is a species of tree rats but only in the South.

Thanks for the info, here's an update. The problem is around the cupola pop-up in the center of the house. All four corners were chewed through. After inserting steel wool and foaming in, I took another look within the attic. The black felt paper right at the edge of the cupola was sagging, with numerous droppings on top. Note that the roof is metal on top of shake shingles. So now I don't know whether in-between the felt paper is where they were exploring after getting in through the corner, or whether that's a new entry point.

So I inserted steel wool and foamed in between there as well. I also screwed in a long steel plate between the wooden slats over the sagging and torn felt paper on one side. The other side only allows a couple of feet leeway due to it being vaulted. Today I crouched under like a mechanic looking under a car and saw that it was torn too, with some old insulation hanging out that was gray. Except when I took a closer look it was a dead rat. Not sure when it died, maybe when I foamed in between the felt paper from the other day. So now I'm going to screw another long steel sheet to fix that hole and sagging.

I just hope they are not living in-between the metal roof and the shake shingles, because they could then chew through the felt paper below since it's not solid plywood below that.

I ordered a ratzapper which should arrive in a few days, and after catching the rest I suppose I'll just wait a few months to see if that stopped them for good. If not, I guess I'll have to replace the whole roof which is expensive.

I know it's rats, as roof rats are common around here. And I have a few fruit trees near the house, and ivy surrounding as well, so fertile ground for them unfortunately. If only I could charge them rent, that'd pay for the roof at least.

Another update: a contractor took a look and thinks they're getting in at the end of the house and then going in-between the roofing paper and shingles all the way to the center. Because there is no plywood underneath, they can chew through the roofing paper and drop down at will, so unfortunately I'm going to get the roof replaced.