$T^*$: $W$ $\to$ $V$.
$\langle$v$,$T*w$\rangle$ $=$ $\langle$$Tv,w$$\rangle$
$=$ $\langle$$\langle$$v$,$u$$\rangle$$x$,$w$$\rangle$
$=$ $\langle$$v$,$u$$\rangle$$\langle$$x$,$w$$\rangle$
$=$ $\langle$$v$,$\langle$$w$,$x$$\rangle$$u$$\rangle$
I'm not sure how he got the last two equalities here. I'm thinking the second last equality comes from the fact that an inner product is a number so if we let $c$ $=$ $\langle$$v$,$u$$\rangle$, then $\langle$$cx$,$w$$\rangle$ $=$ $c$$\langle$$x$,$v$$\rangle$ $=$ $\langle$$v$,$u$$\rangle$$\langle$$x$,$w$$\rangle$. And likewise, for the last equality, we can use the same reasoning along with symmetry of the inner product to obtain the same answer.