How should I understand fundamental class of diagonal of product space

Question

I have some difficulty understanding the fundamental class of diagonal of product space of compact oriented manifold $\Delta \subset M \times M$. For example, when $M = S^1$, the generator of $H^1(S^1 \times S^1)$ is $a \times b$ and $b \times a$ by Kunneth formula. But how can one conclude that the fundamental class of $\Delta \subset S^1 \times S^1$ is $a \times b + b \times a$?