when you play mystic spindles, you can earn "intelligence experience points" and when you have enough you can use them to "buy" intelligence points for your pet. the cost of each intelligence point increases as your pet's intelligence gets higher - i believe it's (N*10)+10, where N represents your pet's intelligence. so if your pet has 0 int, one point will cost 10 exp, and if your pet has 15 int, one point will cost 160 exp.
there are also some stat boosters that increase intelligence.