Understanding $\epsilon$ transitions on NFA
$\epsilon$ is an empty string just like "". It is not a symbol of the alphabet. It simply means (in the NFA in your picture) that if you are on state $q_1$, without consuming any symbol you can move to state $q_3$ but you are not obliged to do so. but if you are on $q_1$ and you read $b$, then you must go to $q_2$.

Note that a word is accepted in a NFA if there exists a path from the initial state to an accepting state labelled by the symbols of this words. if should be rejected if no such a path exists.

When $a$ is entered then the possible path is move to $q_3$ without consuming anything and then consume $a$. This lead to the accepting state $q_1$.