The ability to level up in Elden Ring is something you have to actually unlock. It occurs very early on, but you could be wondering why you can't for the first hour or so. Essentially, you need to be visited by Melina at a Site of Grace. She is the NPC you give Runes to in exchange for character upgrades.
Melina will introduce herself after you have interacted with and rested at your third Site of Grace. It doesn't have to be any three in particular: you just have to rest at any random ones and open up fast travel back to them. After resting at the third Site of Grace, a cutscene will automatically play and Melina will visit you. For most people, this will happen at the Gatefront.
Now you've unlocked the ability, you can select the Level Up option after resting at any Site of Grace to begin spending the Runes you've got. Levelling up will be fairly cheap at the start of the game, but with the cost increasing every single time you do get a new stat point, you'll eventually need Runes in the five figures. If you need more Runes, then refer to Best Locations to Farm Runes to Level Up Fast.