Fisgava comes from the verb fisgar, which literally means to 'hook' or 'catch with a hook', often used in the context of fishing.
In this song, the artist uses it metaphorically to describe being 'hooked' or 'reeled in' by someone's charm. He sings, "Que ninguém mais me fisgava" (That no one else would hook me), expressing his previous resolve not to fall in love again, only to have it broken by a new person. It's a vivid and engaging way to talk about falling for someone.
This song is about giving up on love, and then falling into it anyway.
The singer opens with a vow: never get attached again, never let anyone hook him. He had convinced himself love was over for him, maybe even that it had "forgotten to be born" in his life. Then someone comes along and undoes all of it, one kiss at a time. His guarded heart thaws, and he ends up asking her to stay for the rest of his life.
The title, "Até Conhecer Você" ("Until I Met You"), is also the song's turning point, repeated again and again in the chorus. Everything before meeting her was cynicism and self-protection. Everything after is admitting he was wrong about love, and about himself.