Whenever I watch YouTubers play League they have this gray text appearing telling them how much EXP they gain when they kill a creep. How do I enable that?
Open up your settings either in-game by pressing Esc, or by pressing the small cog wheel in the upper-right corner of the League of Legends client.
Then, navigate to the Interface section on the left side of the Settings window. In the Combat Text subcategory, toggle on Experience Gains.
Here's a picture that indicates where this option is when out of game:
When in-game, it's in the same categories, the display is just a bit different. Here's how that looks:
You will now see exp drops appear when you kill a minion/creep, like the following animated GIF showcases: