You level up by earning XP through various methods: harvesting, collecting from Gear and Animals, making recipes, collecting from ribbons, etc. and reaching a certain threshold.
Your level and XP bar are displayed in the top-right part of your screen (Next Level, Your Stats & Rename Farm).
To see how much XP you need to reach the next level, hover over that bar. When you click on the bar, a new window will pop up.
The Next Level tab shows what rewards await you when you reach the next level and what items you will unlock.
Every time you level up, you will receive 1-4 Ranch Cash (depending on your level) and some Coins. The Power bar will also refill to the maximum of 30 Power each time you level up after level 10 — when the Farm Kitchen becomes available. The Fuel bar also refills to its maximum of 10 Fuel when you level up.
The Stats tab displays your records and some activity stats that are updated at the end of the day, Your Activity Badges, your Kitchen Rank and the Daily Reward button.
The Rename Farm tab is the place where you can rename your farm and choose your nickname (customize how to be adressed by the in-game characters) for Competitions.