What causes the residual magnetic flux when the current is zero in an inductor hysteresis curve?
The easy answer is that it's the residual magnetism in the core (that graph refers to an inductor with a magnetic core).

It's the level of this residual that makes the difference between magnetically 'soft' materials that are used for inductor cores, where this residual should be as small as possible, ideally zero, and 'magnets', magnetically hard materials that retain a large field in the absence of external field. There's a lot of materials engineering goes making the former softer and the latter harder, annealing, alloying, heat treatment etc etc. 

The Ferromagnetism article in Wikipedia gives a thorough if somewhat dry overview of what's going on.

The problem with your question is the word 'really'. There are several explanations at different levels for what causes magnetic materials to exhibit magnetism in the first place, and residual magnetism as per your question. You might talk about domains, or polarisation, or go all the way down to quantum mechanics, and if we knew what was below that (if anything), then we could go there as well. If you're happy with a reality based on one of those, then that should suffice. However some people want a more philosophical answer, and physics doesn't do that.