The three natural signs in the key signature indicates that the key has changed from A major (three sharps) back to A minor (no sharps or flats). If it had changed from A major to Bb major, the key signature at the double bar would show 3 naturals followed by 2 flats.

If you have the old Judy Collins songbook, take a look at the song "Marat/Sade". At one point (p. 176) the key goes from Fm (equivalent to Ab major, 4 flats) to F#m (equivalent to A major, 3 sharps). The double bar there is followed by a key signature that has four naturals followed by 3 sharps.

Note that the use of natural signs in the key signature only occurs once, at the point of the key change. On each subsequent line, the key signature shows the new key.

Does that help, Joe?

Aloha,
Mark