So when I go to generate Pets i have no trait points, and it won't even let me generate them! Also I am slightly confused on how you make an egg hatch.

To get Trait points, and to be able to generate a pet, you have to buy an essence. They are available on the trade broker and in the diamond Shop

eggs will hatch in time- 72 hours maxumum before they prompt you to help them hatch when you go to view their page (an egg symbol appears in the top right of their stall in the hatchery to show this)
The time an egg has left until hatching is reduced by them getting nurtures from people- five nurtures takes off an hour of time.

The generator is like a preview machine- you can play and design, save those designs.
To use the generator to generate a pet, however, you- as mentioned by Hopeless- need an essence. Regular essences give you 2tp (2carry or 1vis), Enhanced and Mystical give 4tp.

To select an essence you have, you click the 'load essence' text that's top-middle of the generator, select the rarity of your essence in the popup that appears, then scroll and click when you locate the one you want to use. Then just hit 'load essence' (if you were looking at a design on a different species from the essence selected, it'll swap to the essence's species, clearing the traits but leaving the colours intact)