use water and fertilizer? what exactly is it for?

Water is used to rise the grass quality in a pasture and fertlizer helps to rise the hay quality.

From the game guide:

<font class="mainline">You can purchase Pastures in the Grow section with Eden Bucks or USD. The benefit of putting your horses in Pastures is that they will require less hay to be fed. When keeping horses in pastures, your pasture will have a Pasture Quality trait. The highest quality is Lush Pasture, the lowest is Barren Pasture. Pastures will degrade over time from highest to lowest, and the speed at which it degrades is based upon how many horses you have in the Pasture. High quality pastures provide more feed to horses and so less hay is needed. Lush pastures require zero hay supplementation, while barren pastures require full hay supplementation.

If you move your horses out of a pasture, the pasture will grow back it's grass, all the way to Lush Quality. However, in the Winter grass does not grow. So it will take longer to grow back during the winter months.

If you put Mares and Stallions of breeding age in a pasture together, they will try to mate. Please read about Pasture Breeding in the Breeding section of this guide for more information.

<font class="mainline">If you own a pasture you may use it for growing hay. The pasture should have no horses in it. When Spring arrives there will be an option on the pasture page to sow hay. Hay takes 12 game days to grow before it may be harvested. Hay pastures have a concept called Crop Quality which indicates the strength of the harvest. It is based on the weather. Rainy days will cause the Hay to grow and strengthen the harvest. Hot days will weaken the hay. There is a numerical value on the Hay pasture's page representing it. Neutral is 10. Once your Hay is ready to be harvested, the amount of hay you harvest is based on the Crop Quality. If you don't harvest your hay by Winter, it will die and revert to a horse pasture.

The number of square/round bales you'll harvest will vary based on your Crop Quality (affected by weather conditions)

When you harvest hay it will automatically be added to your Hay Stock Pile. You may choose to sell Hay in the store. Your best bet for selling hay is to work out the math and price it below the Store Stock Prices.