Kronecker studied philosophy in his youth, in particular the doctrines of Hegel. I think Hegel identified the absolute giest (mind/spirit), an actual infinite reached the final resolution of his dialectic, as God. Hegels doctrines were wildly popular. He had declared his faith in a 'concrete God'. I assume his phenomenology of spirit was to construct him, rather I should say conceptualise him. In the naive way we talk about the qualities of God, like goodness and mercy.
Kronecker converted from Jewish faith to Christianity. He was opposed to non-constructive proofs, he demanded a construction of the actual object. He was in fact against the unrestricted use of the infinite in arguments & that of the use of the excluded middle. In fact one could say he initiated a new school of mathematics - intuitionism.
Cantors theories of the infinite may have appeared to encroach on that territory as Dain points out in his answer; and Kronecker with a new converts zeal, may have attacked him on this very point, hence the call of impiety, an attack on his faith and, since he didn't believe in infinities or the reductio, his charge of the socratic corruption of youth away from truth, was him being simply consistent with his position.
But ironically, Cantor himself worried about that too, he had mystical tendencies towards the absolute infinite (perhaps he too had read Hegels doctrines). Aristotle, believed in a potential infinity, but not an actual one. Cantors theories of the infinite, far from disproving this claim, only affirmed them. It kind of leads one to suspect that Kronecker never spent the time to understand Cantors theories.
But they needn't have worried, current set theorists have considered large cardinal axioms that go far, far beyond the kind of infinities envisaged by Cantor. And still there is territory far, far beyond what they have currently surveyed. In a way, considering the territory yet to cover, it's as almost as though we've never even come out of the safe shelter of the finite.
I'm sure, but without proof, that Cantor was aware of this, hence his mystic leanings towards the absolute infinite, an idea, which had been possibly conditioned by Hegel, and perhaps platonism (I don't know if he was that way inclined). One would suspect so, since his emphasis on consistency being the sole judgement of mathematical character.
Poincare may have said that 'there is no absolute infinity', and that the Cantorians had forgotten this, but I very much doubt that Cantor held the same view. He shouldn't be judged on the polemics & antics of his band of followers.
(If one views Kronecker as the father of intuitionism arising in opposition to Cantors formalism; one can view todays multiverse of sets now only dimly seen, as through a mist, the synthesis of these two contradictory positions. A rather nice example of a Hegelian dialectical movement - if it works).