What is the need to level up in the game?
As it is with any other game, leveling up is synonymous to progress. Although you can technically play the game without having to fixate leveling up your character level (which should do just fine), it is important to understand that every time you level up in this game basically means you are unlocking more of its content (more animal friends to meet, more Furniture and stuff). Below are the benefits that you gain every time you level up in the game, or reach a certain level.
More Animal Friends becomes available
Making friends with animals is probably your bread and butter in this game. As you have probably guessed by now, getting buddies with these animals will, in return, provide you the basic stuff to build something in the game. Not limited to this, they themselves technically unlock the variety of items to Craft in the game. As you reach specific levels in the game, you get to unlock more animals to be friends with.
Level Brackets and Unlocked Animals (to be friends with):
Game Start - Apollo, Filbert, Goldie, Jay, Rosie
Level 3 - Beau, Butch, Cherry, Eloise, Lily, Tex, Bunnie
Level 9 - Apple, Bud, Carrie, Chrissy, Ketchup, Kid, Cat, Maggie, Sandy, Tad, Punchy
Level 19 onwards - Agnes, Stella, Cheri, Alfonso, Angus, Bitty, Charlise, Fauna, Flip, Hamlet, Hopkins, Kyle, Moe, Peanut, Peewee, Rex, Roald, Roscoe
You get to invite more in your camp site
Inviting (animal) friends to stay at you camp has a different benefit when compared to those that are roaming free outside. For one thing, those that are staying at your camp will often times bring you gifts (Crafting materials). On top of that, Animal friends staying on your camp, tend to increase their relationship level (with you) faster (depending on the Furniture and Amenities you have). Also, you get to unlock the cap Friendship Levels of these animals (when they reach level 7 which is the default Cap Level) by asking them to stay in your camp, and building something that they would specifically request. Basically, you get to invite more animals (to stay at your camp at a time) as you level up in the game.
New stuff to Craft becomes available
As we have stated above, more animal friends technically means more stuff to unlock -- as they basically dictate which raw items you get to acquire while playing. More assortments of raw building materials generally makes the variety of stuff you get to produce in the game. If so, your character level pretty much dictates which animals you get to be friends with, which equates to the scope of raw materials you can currently get a hold of.