Not sure how cold it gets where you are and how much heat you're adding in, but I'm guessing the reason for the mesh instead of fascia is to give some ventilation to the roof
Honestly it probably doesn't matter if you close it up, sacrifice a little rot and ice buildup for your comfort, you said yourself this is just some barn structure, you're not going to wreck it overnight
I'm in Canada, up here we like ventilation in our attic spaces to take away the condensation as warm humid air from inside cools where it meets cold outside air, so you'd normally want a vented soffit with a nice air path leading out the top of your roof, minimizes ice buildup.
For this, you have vaulted ceilings and it looks like basically no space for insulation+ventilation between the inside ceilings and the roof deck, so without knowing your climate or how things are built there I'm betting they saw moisture building up in that eave and removed the fascia to let it breathe
Close it up if you want, I wouldn't bother foaming it, and if you notice things are too wet just take the boards off. Again, you won't wreck it overnight.