I received this achievement from a scratchcard

Level Up! Win stat upgrades from the scratchcards! You have this achievement!

What exactly does this mean - a level for pets ?

It just means that one of the stats (strength, health, level, etc) went up when you scratched your card.

Pets can train stats and level at the training centre in riverside. Mostly, this in only important for achievements and/or battling.

Thanks for that. My pet practically lives in the training centre, and I didn't notice when I scratched the card.