If multiple +1/+1 counters are placed on Fathom Mage simultaneously, its last ability will trigger once for each of those counters.

If Fathom Mage enters with a +1/+1 counter on it (perhaps due to Master Biomancer), its last ability will trigger once for each +1/+1 counter it entered with.

Fathom Mage's last ability will trigger whenever any +1/+1 counter is placed on it, not just ones due to the evolve ability.