Why can PDAs only write one symbol to the stack according to this definition?
The addition of $\epsilon$ to the alphabets allows the pop and push operations. For example, if you want to push the letter $a$ in state $q$, without reading anything from the input, you can have the transition $\delta(q,\epsilon,\epsilon)=\{(s,a)\}$, where $s\in Q$.

And if you want to pop $a$ without reading anything from the input, you can have $\delta(q,\epsilon,a)=\{(s,\epsilon)\}$.

Of course you can combine the two for a simultaneous pop/push, and you can do that while reading from the input.