There's really not much to say about the nursery. The nursery is the place where you sell your plants. You can purchase upgrades for the nursery from the supplies menu. Some of the supplies have an effect when you click on them; the turtle, for example, will squirt water. I also think (but I don't have proof for this theory) that when you upgrade your nursery, people will be more willing to buy plants.
When you have very few upgrades, you will see this image in your guests' thoughts:
When you start to get a lot of upgrades, this thought will replace the above one:
When you first start the game, your nursery will look like this:
The completed nursery will look like this (I blacked out the magic plants so you won't see what they are without meaning to).