does the +1 power also increase the health of the creature or does it stay at its base?

There isn't health in this game. Only power.

A creature dies when it has damage on it equal to or greater than it's power. So increasing it's power means it can take more damage.

"There is no health... Only POWER!"

Now I want that on a T-shirt.

Woah, That means cards like Zyzzix the Many can be drowning in - and + counters.