What can I do with trainer levels?

Solution 1:

There are different rewards as you level up:

• As your level increases, so will the CP of wild Pokémon (which means you can capture stronger Pokémon) and you can find more species.

• Your level determines the CP cap for your Pokémon, so higher player levels mean that you can power up your Pokémon to higher CP.

• At level 5 you can start challenging gyms and participate in battles.

• At level 20, you receive Nanab Berries instead of Razz Berries. Then every odd level after 20 it will alternate between Nanab and Pinap Berries as a reward. source

• Each level up grants item rewards, here is the list:

Level 1: (Starting Level)

Level 2: Poké Ball x10

Level 3: Poké Ball x15

Level 4: Poké Ball x15

Level 5: Poké Ball x20; Potion x10; Revive x10; Incense x1

Level 6: Poké Ball x15; Potion x10; Revive x5; Egg Incubator x1

Level 7: Poké Ball x15; Potion x10; Revive x5; Incense x1

Level 8: Poké Ball x15; Potion x10; Revive x5; Razz Berry x10; Lure Module x1

Level 9: Poké Ball x15; Potion x10; Revive x5; Razz Berry x3; Lucky Egg x1

Level 10: Poké Ball x20; Super Potion x20; Revive x10; Razz Berry x10; Incense x1; Lucky Egg x1; Egg Incubator x1; Lure Module x1

Level 11: Poké Ball x15; Super Potion x10; Revive x3; Razz Berry x3

Level 12: Great Ball x20; Super Potion x10; Revive x3; Razz Berry x 3

Level 13: Great Ball x10; Super Potion x10; Revive x3; Razz Berry x3

Level 14: Great Ball x10; Super Potion x10; Revive x3; Razz Berry x3

Level 15: Great Ball x15; Hyper Potion x20; Revive x10; Razz Berry x10; Incense x1; Lucky Egg x1; Egg Incubator x1; Lure Module x1

Level 16: Great Ball x10; Hyper Potion x10; Revive x5; Razz Berry x5

Level 17: Great Ball x10; Hyper Potion x10; Revive x5; Razz Berry x5

Level 18: Great Ball x10; Hyper Potion x10; Revive x5; Razz Berry x5

Level 19: Great Ball x15; Hyper Potion x10; Revive x5; Razz Berry x5

Level 20: Ultra Ball x20; Hyper Potion x20; Revive x20; Nanab Berry x20; Incense x2; Lucky Egg x2; Egg Incubator x2; Lure Module x2

Level 21: Ultra Ball x10; Hyper Potion x10; Revive x10; Pinap Berry x10

Level 22: Ultra Ball x10; Hyper Potion x10; Revive x10; Razz Berry x10

Level 23: Ultra Ball x10; Hyper Potion x10; Revive x10; Nanab Berry x10

Level 24: Ultra Ball x15; Hyper Potion x10; Revive x10; Razz Berry x10

Level 25: Ultra Ball x25; Max Potion x20; Revive x15; Pinap Berry x15; Incense x1; Lucky Egg x1; Egg Incubator x1; Lure Module x1

Level 26: Ultra Ball x10; Max Potion x15; Revive x10; Razz Berry x15

Level 27: Ultra Ball x10; Max Potion x15; Revive x10; Nanab Berry x15

Level 28: Ultra Ball x10; Max Potion x15; Revive x10; Razz Berry x15

Level 29: Ultra Ball x10; Max Potion x15; Revive x10; Pinap Berry x15

Level 30: Ultra Ball x30; Max Potion x20; Max Revive x20; Razz Berry x20; Incense x3; Lucky Egg x3; Egg Incubator x3; Lure Module x3

Level 31: Ultra Ball x10; Max Potion x15; Max Revive x10; Nanab Berry x15

Level 32: Ultra Ball x10; Max Potion x15; Max Revive x10; Razz Berry x15

Level 33: Ultra Ball x10; Max Potion x15; Max Revive x10; Pinap Berry x15

Level 34: Ultra Ball x10; Max Potion x15; Max Revive x10; Razz Berry x15

Level 35: Ultra Ball x30; Max Potion x20; Max Revive x20; Nanab Berry x20; Incense x2; Lucky Egg x1; Egg Incubator x1; Lure Module x1

Level 36: Ultra Ball x20; Max Potion x20; Max Revive x10; Razz Berry x20

Level 37: Ultra Ball x20; Max Potion x20; Max Revive x10; Pinap Berry x20

Level 38: Ultra Ball x20; Max Potion x20; Max Revive x10; Razz Berry x20

Level 39: Ultra Ball x20; Max Potion x20; Max Revive x10; Nanab Berry x20

Level 40: Ultra Ball x40; Max Potion x40; Max Revive x40; Razz Berry x40; Incense x4; Lucky Egg x4; Egg Incubator x4; Lure Module x4

Level 40 is the maximum level

My source seems to be updated often (yesterday it only had reward info up to level 12)

Edit: Another source

Solution 2:

Leveling up unlocks a very useful feature, Pokestop Submission. It was available only for level 40 players, but in late 2020 the required level was lowered to level 38!

Trainers who've reached level 38 can help grow the number of PokéStops in their community by nominating eligible objects and locations in Niantic Wayfarer.

Source, Niantic Support.

You also have some in-game benefits.
As your level increases, you will discover more Pokémon and face Pokémon with stronger CP in the wild. You'll also get various perks and unlock new items.

Here's a nice table that summarizes the rewards through level 20:

UPDATE:

With Go Beyond update, level cap increased from the previous level 40 to level 50.

As well as meeting a certain threshold of XP, now you also need to complete certain challenges - named Level Up Research - in order to progress to the next level:

You can find further info and relative rewards on Serebii.

Solution 3:

• Level 5: you can join teams and get to do gym battles.

• Level 6: you can get a new type of Incubator with 3 uses, allowing us to incubate and hatch several eggs at the same time.

• Level 8: you can get razzberries, which make the next ball you throw have an increased chance of cathing the pokemon.

I've not found any other information regarding leveling up, but I will update this as new information is presented.