Non Deterministic PDA accepted language not clear
The PDA you're linking to is non-deterministic; indeed, the state you are confused about has two out-edges labeled $a$. For the PDA to accept its string, it's enough for there to be one accepting computation – what you refer to inaccurately as "human interference" [intervention?]. Regarding the acceptance condition of the PDA, figure out what acceptance condition makes this PDA accept the advertised language, and this is the accepting condition that you should have in mind.