In the diagram position, White has an extra bishop, and as compensation Black has both two extra pawns and, as you mention, prospects for an attack based on the fact that the white king is out in the open on the third rank. If White can secure the king, though, then his extra bishop will mean more in the long run than the two pawns, and White will be doing much better.
With his last move,
9. a3, White is threatening to do precisely that, as he as opened a retreat path for his king to make use of (once he takes care of the fact that his a4 knight would be hanging at the moment). Thus if Black were to make some nothing move, e.g.
9. ... h6?, then play could proceed with something like
10. Nc3 Qf5 11. Ka2 (though here White doesn't even need to be in a hurry rushing his king back, and could do something more productive instead) and White is just cruising.
In broad strategic terms, Black must act quickly to take advantage of the dynamic factors currently in his favor (White's exposed king and lack of development), or White's long-term static plus (extra material) will win the day.
9. ... Qxa4+ does exactly that, giving up (much) more material, but in return forcibly drawing the white king further out into the open, where it turns out Black has enough firepower in reserve to draw with a perpetual check.
One could try other ninth moves for Black of course, but again the point is that matters are urgent: something must be done now about White's consolidating idea of
Nc3 -> Ka2, and there's only so many moves that have potential.
9. ... d4 keeps the knight from c3, but it also removes the attack on the knight, so White can play
10. Ka2 immediately. Instead
9. ... Be6 is a nice thought, with the idea of discovered check by
10. ... d4+ keeping the white king out in the open; but unfortunately
10. d4! in reply stops that idea and leaves White firmly in control. (Incidentally, if White had gone ahead and played
9. d4 instead of his
9. a3, he would have already killed off the attack and had a big advantage, but then we wouldn't have the Immortal Draw.)
I could go on, but I think the above spells out the exigency of Black's situation. If you're still curious from here why
9. ... Qxa4+ is the way to go, the best thing to do is probably some concrete analysis, toying around with possible variations starting from the diagram position above (with a trusty engine looking over your shoulder).