Acceptance of empty string by nondeterministic automaton
You're confusing the concept of the empty string&nbsp;$\varepsilon$ with the concept of an $\varepsilon$-transition.

The empty string&nbsp;$\varepsilon$ is just the string with no characters.  The presence of a $b$-transition from $q_1$ to&nbsp;$q_2$ means that the automaton can move between those two states by reading character&nbsp;$b$ but the presence of an $\varepsilon$-transition from $q_1$ to&nbsp;$q_3$ doesn't mean that it moves by reading the character&nbsp;$\varepsilon$. Rather, it means that the automaton can (but is not forced to) move from $q_1$ to&nbsp;$q_3$ without reading any input at all.

So, when this automaton has no input, it can either remain in state&nbsp;$q_1$, which is accepting, or move to&nbsp;$q_3$, which isn't.  Since it can reach an accepting state, it accepts.