Choosing one move from the two moves From the given position, is it better to take with rook or the a-pawn and why ?

Black to move:

The engine suggested the move axb4, why?

There must be some strategical or tactical point (ignore the arrows in the diagram).
After axb4 the a2-pawn is a fixed weakness that can easily be further attacked with Ra8. Defending this pawn will force whites heavy pieces into passive positions. 

Taking with the queen or the rook on b4 will allow Rb1 with a rook exchange. Even if this rook exchange happens on b4 and therefore creates the same pawn structure as after an immediate axb4, the defence of a weak a-pawn will be much easier without the rooks. In fact white might be able to exchange the a-pawn for the b-pawn relatively quickly.