FWIW, looks like it is a common expression, with a figurative sense, something like "why this shows up here?" See: http://www.accademiadellacrusca.it/it/lingua-italiana/consulenza-linguistica/domande-risposte/centra-centra
Think it this way:
Both are idioms, start with the English one: "What have to do with it? " is "Why is it involved with (X)?" (included in activity X). similar to "Why is it in the picture? ==> why did it enter in the picture? (figure for activity)
Then the Italian one: "Questo que c'entra?" roughly "(And) this that is entering (in some activity), (Why?)?"
So both idioms are questioning about the relationship with something that seems out of place, Just using different figures.
Questo = this
che = that
c'entra = ci+entra = to enter into / to get involved
This that 'is entered '/'gets involved'? ~ What has this to do whit it?
Entrarci is a combination of entrare to enter and the reflexive pronoun ci, - literally meaning to enter into but it is also used to mean to get involved.
C'entra is the third person singular form of entrarci, and it has the same meaning.
For more details on how to use it check thelocal.it
They actually are explicable, if you know the history behind it and sort of take a mental step back and look at the general sense.
Frex., "something fishy" comes from smelling a rotting fish. That's usually bad (unless you're fertilizing your garden, and even then it's not pleasant) so "I smell something fishy" means "I sense something bad/wrong." It DOES make sense, it just uses a different example/expression to communicate the thought than a foreign speaker might be used to.
This is the closest thing to an explanation that I could find: http://forum.wordreference.com/threads/ma-che-centra.116772/