most anthuria are epiphytic, semi-epiphytic or lithophytic. Their usual home in the wild may be as simple as the crux between tree branches that they can attach to and sit in a cache of leaf litter.

When put into soil or the ground, they will almost always send roots up into the air, because that is their epiphytic nature.

The best way to repot an anthurium is into a large basket, like a mesh basket or a wood slat basket lined with mesh, filled with a mix of organic stuff like bark, peanut shells, some humus, perlite, aliflor, etc. The roots are looking for air when they come out of the dirt like yours are. You can always set the basket into a large cachepot.

I'd second most of that. It's true, however, that even here in Brazil, Anthuriums are often grown as pot plants and they don't mind this. Some even say they flower better when pot-bound. My mother-in-law has some spectacular plants that are only in 10" pots. You must keep them warm , however.

My wife received her's as a gift in the hospital . It was in a small pot, which it quickly outgrew.
I transplanted it to a larger and it never seemed to slow down.
My question: How and when should it be fertilized?