If I understand correctly, a strategy for a two-player game (for either player) is a function from $\omega^{<\omega}$ (i.e. the set of all finite sequences of natural numbers) to $\omega$. Jech claims that the cardinality of the set of all strategies is $2^{\aleph_0}$.
I suspect the proof would go something as follows?
First, we show the cardinality of $\omega^{<\omega}=\omega$. At least $\omega=|\omega^{<\omega}|$, since $|\omega^1|=|\omega|\subset|\omega^{<\omega}|$. Also, $\omega^\omega=\omega$ (here I am talking about ordinal exponentiation...). It then suffices to show the set of all functions from $\omega\to\omega$ has cardinality $2^{\aleph_0}$, but this is a standard result in set theory.
Is this about right? Apologies in advance for conflating $\omega$ and $\aleph_0$. I've seen so many texts abuse this notation that I've become accustomed to it myself.