Vronsky has learned, earlier that day, that Anna is pregnant. It's obvious that the horse's death is tied to this revelation. The pregnancy, which forces her husband to acknowledge the affair, leads to the breakup of their marriage and further tragedy. Vronsky knows that he caused the horse's injury, and yet he will not admit the reason. His agitation must have affected his skill as a jockey, but his nature is to avoid responsibility and self-examination. The only other explanation is bad luck. Vronsky seems to lead a charmed life, but that has just changed.
Tolstoy notes, immediately after the horse's injury, that Vronsky will remember this as the darkest moment of his life. Another person might remember it as the day he learned he'd be having a daughter, but Vronsky is selfish. To him Anna is just another conquest. He does his duty by her, as he understands it, but no more.
Frou-Frou's death, from this vantage, is a symbol of the change in fortunes for Vronsky and Anna, not where those fortunes will lead. Anna's death has already been foreshadowed. Until this point, both characters could have gone back to their previous lives without serious repercussion.
According to the text, "the horse's back was broken and she had to be killed" (Nathan Dole translation, Internet Archive, page 261). He has gone from his initial rage to cradling her head before the medical staff arrive. It's not explicitly stated that Vronsky shoots Frou-Frou himself, which one would expect if he had.