When you're loaded into a Formicarium, in the top left is a menu that allows you to launch levels. It's the second hexagonal-shaped button - I've highlighted it for you here:
From here, choose a level. To start off with you'll only be able to select 1.1 New Home, 1.2 Subjugation or Formicarium Challenge 1. Select New Home, choose a difficulty for it, and then play the level:
Upon completing the level you will be given rewards of territory (allowing you to dig out more tiles in your formicarium), food (which is dropped on the surface of the formicarium - you'll need to excavate the nest entrance tile to the far north to activate it) and royal jelly, 48 in the example above, which is used to unlock new ants and upgrades for the, from the royal jelly tech tree menu. That menu is the hex with the DNA picture above the level launch menu.
This is all explained in the popups, so make sure you're not ignoring or closing them before you know what information they're giving out: