I'm confused by the following paragraph:
I don't see why $g\cdot W$ depends only on the left coset $gH$. What does he mean precisely by that? Why is it true that $gh\cdot W = g\cdot(h\cdot W) = g\cdot W$? It's only true that $gh\cdot W = g\cdot(h\cdot W) \subset g\cdot W$, how is the reverse inclusion true?
Secondly, why is the sum direct in $$\bigoplus_{\sigma\in G/H} \sigma\cdot W$$
About the first question: $g\cdot W$ depends only on the left coset because $W$ is $H$ invariant. You have that if $gH = g'H$, then $g\cdot W = g'\cdot W$ exactly because $$ h\cdot W = \{h\cdot w : w\in W\} = W. $$ (Remember that $W$ is a representation of $H$.)
About the second question. This is just the definition. That is, $V$ is the induced representation exactly when each element in $V$ can be written as a unique sum of elements $\sigma w$, for $\sigma\in G / H$. That is, for all $v\in V$ you can write $$ v = \sigma w_1 +\dots + \sigma_n w_n $$ for unique $w_i\in W$ and $\sigma_i \in G / H$.