Q:

Why is the bridge index of the trefoil equal to 2?

It seems to me, all three 3 bridges are needed?

A:

The bridge index of a knot $K$ is the minimum number of local maxima (or equivalently, minimum number of local minima) of the height function on all possible knot diagrams of $K$. So from the standard picture of the trefoil, you can see that its bridge index is at most $2$. It is clear that the only knot with bridge index $1$ is the unknot, so in fact the bridge index of the trefoil $T_{2,3}$ is $2$. Here's a picture of $T_{2,3}$ with its standard $2$-bridge presentation:

