I was black. I had +2 material advantage, but after Qxb2, white did Qxc6+ and won the game.

Best move in this position is Qb7, but I always go for the material advantage and don't notice that kind of stuff.

So when exactly will this guy get a chatban?

Explain in detail why Qb7 is a much better option than Qxb2 in this position.

I ran the position through Stockfish at depth 26-28 to confirm it with concrete lines. The verdict is decisive:

Engine evaluation:

13...Qb7 -1.39 (uncomfortable but playable position)
13...Qxb2 -7.66 (practically lost)

The problem with Qxb2

At first glance, Qxb2 looks like it wins a pawn and creates a fork: the queen on b2 simultaneously attacks the rook on a1 and the knight on c3, both on the a1-h8 diagonal. It looks like a clean tactical shot.

The problem is that by leaving b6, the queen stops defending the c6 pawn (which was protected horizontally by the queen on the sixth rank). And that's the key point: your king is still on e8, uncastled, and the d7 square is empty. That allows:

14.Qxc6+! — a check that redirects all your attention away from the "fork" you just created.

You only have one reasonable response: 14...Nd7 (blocking with the knight, since the king has no good squares).

And now comes the second blow: 15.Rab1!, attacking your queen on b2, which has no safe retreat square. You're forced into 15...Qxb1 16.Rxb1, trading your queen for a single rook.

Result: you've given up your queen for a rook (a material difference of about 4 points), with your king still stuck in the center, uncastled. The engine even finds alternative lines for Black (15...O-O-O, 15...f6, 15...Qb5, 15...Rd8) that are even worse, with one line leading to forced mate.

Why Qb7 is sound

Qb7 maintains the defense of c6 (b7-c6 are diagonally adjacent), doesn't commit prematurely, and lets you develop normally (...e6, ...Be7, castling). The engine continues with something like 14.Rad1 e6 15.Ne4 Be7 16.Nc5 Qc8 17.Bd4 — White keeps a slight initiative, but nothing catastrophic. It's the difference between being somewhat worse and being practically lost.

The underlying tactical lesson: before rushing to grab a "free" pawn with a check-included fork, you need to check what squares/pieces your queen stops defending when it moves — in this case, the intermediate check Qxc6+ (possible only because your king is still on e8) completely breaks the operation.

Can I throw a table for raging too hard at an AI message?

To put it in a single line, you just missed Qxc6+ (A fork), and instead of defending c6 with Qb7, you greedily grabbed the b2 pawn and lost.

AI is pretty much useless, as reading the entire thing is simply a time waste.

Qb7 is the only option to protect the weak c6 pawn. So in case the c6 pawn is not defended, Qxc6 and then, the white queen can take your rook!