How mandatory is leveling up in League of Legends?

There are 2 kinds of "Leveling" in LoL:

Leveling your summoner
In League you take the role of a "Summoner". Basically every character you actually play is summoned by your summoner. Increasing your summoners level allows you to unlock more Configuration outside the actualy game such as Summoner spells, Masteries and Runes/Rune Slots. You will increase your summoners level by Playing games (Winning will give you more experience than losing) and you can also buy EXP Boosts for Riot Points (Real money) to speed it up. You will gain the following Boni from summoners levels:

• A new skill point every Level
• A new Rune slot every Level
• Access to stronger runes at level 10 and 20 (Don't buy runes before level 20)
• A new summoner spell every level until level 12
• Access to new game modes every few levels (Not sure right now when exactly but I'll add it later)
• For reaching some levels you will gain a small amount of free riot points

Basically your summoners level is a method that's used for matchmaking alongside with your win/loss ratio. And while It certainly does have an Impact on the game itself (A level 10 player will be a bit weaker than a level 30 with masteries/runes), It's not as extreme as in WoW or other MMOs (A level 85 will get 1shotted by a level 90 there).

Leveling your Champion
The other form of leveling is Leveling your In game champion. Each time you start a game your champion will be level 1 in summoners rift or level 3 in dominion and ARAM. This cannot be modified since the only things that transfer from Client to Game are Summonerspells, Runes and Masteries.

Your Champion can be leveled up to level 18 and He will gain Base Stats per level such as:

• Attack Damage - Enhances you Physical/Autoattack Power
• Attack Speed - Well.. Name says it all.. You hit faster
• Magic Resistance - Only for meele characters with few exceptions, Resistance against Magical Damage
• Armor - Thresh is the only champion that doesn't gain armor per level. Resistance against Physical Damage
• Health - More Life (duuuh...)
• Mana - Allows you to cast spells, not every champion has mana as a resource
• Health regeneration - See name
• Mana regeneration - See name

Stats that won't increase on level-up are:

• Colldown Reduction - Lower cooldown, Caps at 40%
• Ability Power - Increases spell damage for AP-Scaling Abilities
• Lifesteal - Basic attacks return health
• Spell Vamp - Spells return health
• Critical Chance - %Chance of hitting for 200% damage
• Critical Damage - %Increased damage of Critical strikes
• Energy Regeneration - Regeneration rate of champions who use Energy as a primary resource

There are also a few additional things that champions gain per level such as more range (Tristana) and Higher Shielding amount (Yasuo,Mordekaiser). However the most important thing you gain from Champion levels are access to skills. For every level you gain you gain another ingame-skillpoint that you can spend on any of your 3 First abilities (sometimes champions automatically start with their ultimate or in the case of Udyr it's unlockable at level 1). And at level 6/11/16 you can increase your Ultimate Skill (True for all cases but udyr).

Different from The summoners level, the in game level has a huge impact on whether or not you can duel a certain champion, since every level increases the effect of a spell by a high amount.

Adding on to RayofCommand's answer, unlocking summoner spells, two free spells that you can choose at the start of every game, is also important. For example, the most versatile and used summoner spell in the game (Flash) isn't unlocked until you reach level 12.

Also, once you get to level 30 you can play ranked matches, in which people play more seriously. Many people simply play normals to have fun and try out new things, while in ranked people usually give their best.