Why is trans-1,2-dimethylcyclohexane an enantiomer?
No, they can't.

A model from any chemical modeling kit will convince you right away.  Or maybe a cheaper alternative made out of chewing gum and matchsticks will suffice.

Or better yet, you may do it in your head. Place the molecule in a horizontal plane (or rather, as close to a plane as possible, since the thing is not planar). Now place your eyes in the same plane, so as to look at the molecule from the side. Walk around until you are facing the bond that connects two methylated carbons in the ring. Look at that bond. Look also at the two methyl groups to the right and to the left from it. You either have the "left one up, right one down" isomer of your molecule, or vice versa. You might think "what the heck, I'll turn it around." No way. You turn each enantiomer around, and it still ends up looking like itself, and not like the other one.

Another way to think of it is via symmetry elements. For a molecule, not being chiral implies having certain kinds of symmetry. Earlier revisions of your question suggest that you were looking for some symmetry plane. Just where it might be? Think of it. And if you don't find that plane after a while, consider the possibility that it might be simply not there (which is true).

Ring inversion is irrelevant altogether. It can be done, but it will switch all equatorial substituents to axial and vice versa, so you won't be able to recognize the same molecule until you invert it again to the original form.

So it goes.