Purple monsters are promising youths (that's why they're level 1 and have great stats) that you can try to capture without lowering their life (because you'd kill them in one hit.
First skill slot for pets is available at level 6 (and no not 5 o_O).
Level your new pet and say goodby to your old one

As you can see he has more strenght and stamina so in the end he'll do more damage and will have more hp

He also have a higher growth (rate) meaning that...well he'll either gain xp or training point quicker ? (this point should be clearly explained in game)