The question gives the PDF using two cases of random variable X. The solution begins with the PDF of X as stated below. I don't understand how they got to that point. I tried to take delta_0 to be 1 if x=0 and 0 if else. And delta_t to be 1 if x=t and 0 if else. But this doesn't fit at all with the computing of the Expected Value.
Your interpretation of $\delta_t$ is correct. It is fine with the computation of the expectation. $$\mathbb E(X)=\sum_{x} x\times f(x)$$ $$=\sum_{x} x\mathbb P(X=x)$$ where sum is over the values taken by random variable $X$. $$\Longrightarrow \mathbb E(X)=0\times \Big(1-\frac{1}{a}\Big)+a\times \frac{1}{a}=1$$