If we can $\epsilon-$cover one set with $\frac{K}{2}$ elements and another with 2 elements why isn't their set product $2^{\frac{K}{2}}$ cover?
Given a cover for $\mathcal{X}$, you seek to "cover" $\mathcal{X}\times\{-1,1\}$.
To do so, you take any set $A$ from the cover of $\mathcal{X}$, and consider the sets $A\times\{1\}$, $A\times\{-1\}$. If your original cover was of size $K$, then this process results in $2K$ sets (the reason you don't end up with $2^K$ sets is that fixing the label for one set does not affect the labeling of another, so there is no multiplication going on).

In a more straightforward approach, let $\mathcal{C}$ be your original $\frac{\gamma}{2}$-cover for $\mathcal{X}$, then a partition $\mathcal{P}$ of $\mathcal{X}\times\{-1,1\}$ such that for all $ p\in\mathcal{P}$ and $(x_1,y_1),(x_2,y_2)\in p$ it holds that $y_1=y_2$ and $||x_1-x_2||\le\gamma$, is given by $\mathcal{C}\times\{-1,1\}$, which is of size $2|\mathcal{C}|$.