How does red wine butter impact the flavor of dishes?
The main point, IMHO, is to make an easy to use and flavorful compound butter. It can be put on top of grilled meats, tossed in with a sautee of meat/veggies/pasta, etc.

I would finely mince some onion or shallot and garlic. Lightly sweat them, then add wine and reduce. Add some fresh herbs and the butter and melt to combine. I then cool this until lightly set, then roll it in a tube in some plastic wrap and freeze it. It can then be sliced off, like tubed cookie dough, and placed on top of grilled/baked meats, etc...

The wine and herbs add a rich, deep flavor that can be used in a variety of ways.