What can I do with a unicorn?

I just got a Unicorn and I don't know how/why I got it.

How did I get it and what can I do with them?

I did get an achievement Achievement unlocked: Unicorn Conspiracy!

And it says, Lift the shroud of the Unicorn Conspiracy!

Your hunters have a chance to find unicorns when you send them out for resources. Unlike the other rare resources they find, furs and ivory, unicorns don't deplete over time. Initially, the only thing you'll be able to do with Unicorns is make Unicorn Pastures. As the other answers have already said, this will give a small amount of Catnip consumption reduction, and also generate additional Unicorns. And just having Unicorns at all gives +10% happiness, which is basically a free +10% production. But, like almost all things in this game, they have more than one use.

The real benefit of Unicorns, though, comes much later. When you research Theology, you will unlock Priests, the Faith resource, and the Religion tab. If you also at this point build a Ziggurat (costs Megaliths, Scaffolds, and Blueprints), a second section opens up. Here, you will gain the ability to Sacrifice Unicorns, returning them to the Unicorn Dimension in exchange for Unicorn Tears. Each sacrifice gives one Tear per Ziggurat owned. There are then a series of buildings available in the Ziggurat section of the Religion tab, each built using Ivory and Tears. These buildings will increase your Unicorn production, and unlock events similar to the Meteors which give free Unicorns and Ivory.

The last building in this section, the Sky Palace, gives a very small chance once built for you to randomly find Alicorns, a variant of Unicorns. Alicorns have the same basic +10% happiness effect, and can be sacrificed for Time Crystals. Time Crystals are required to research Chronophysics, which... doesn't do anything right now. The game's still in development, but ultimately, Time Crystals are meant to be a late-game resource, used in most of the things unlocked by Chronophysics, and the only way to get them will be through the Unicorn-related religion buildings.

You can build a Unicorn Pasture (at the bonfire, if you have two unicorns)! This will reduce your catnip consumption by 0.15% for each (unicorn) pasture you have. And even better, there are coming little unicorns from that pasture (0.001 per pasture per tick), so you have an infinite amount of unicorns! How awesome is that?!

Obtaining a unicorn for the first time provides a permanent +10% to happiness, you'll be able to create Unicorn pastures which will generate more unicorns for you. Unicorns are eventually used for procuring late game resources.

You can obtain Unicorns by sending out hunters, once you have two of them you can start dealing with Unicorn Pastures