Bu akşam misafirlik var; annem teyzemi ve yengemi evde ağırlayacak.

Misafirlik is a noun built from misafir (guest) plus -lik, which often creates a noun meaning a related situation, state, or social event.

So Bu akşam misafirlik var does not literally mean There is guest-ness in a strange way. In natural Turkish, it means something like:

• We’re having guests tonight
• There’s a guest visit tonight
• There’s company coming tonight

If you said Bu akşam misafir var, that would also be understandable, but misafirlik var sounds more like a social occasion involving guests/visiting. It can feel a little broader and more idiomatic.

Var means there is / there are / exists.

In Bu akşam misafirlik var, Turkish is using an existence structure:

• bu akşam = this evening / tonight
• misafirlik = guest visit / having guests
• var = there is

So the structure is literally something like:

• Tonight, there is a guest-visiting situation

This is a very common Turkish pattern. Turkish often expresses ideas with var where English might prefer a different wording.