How to upgrade weapons and especially, why?

I'm relatively new to Team Fortress 2 and I see that all of my weapons have a own level and some of them differ from each other.

Sometimes in matches, I see that some people get an level up on their weapons. How can this be achieved? Is there some achievement which must be reached for a level up on a certain weapon or will they just upgrade if you killed enough bad guys?

What usage can I take from a weapon with a higher level? Does this count in any way? Damage or Reload speed or something like that?

Most likely the 'level up' you're seeing is that the weapon is of Strange Quality. Strange weapons and items will:

• track kills or other various events, (such as kill assists or earned points), and record them as an attribute of the item.
• be awarded 'ranks' as the recorded count increases, with the rank affixed to the beginning of the item's name.

Each time the player achieves a new rank, it is announced to all players on the server with a message in the chat box; A notification pop-up also appears only for the player the strange item belongs to.
_{- TF2 Wiki}

Strange is one of many 'Qualities' that a weapon can have - all of which are purely cosmetic - 'levelling up' a Strange quality weapon does not make that weapon better, it's purely flavour.

The qualities that items (not just weapons) can have are:

• Normal - The stock weapons each class starts with
• Unique - Unlocked from Achievements, Trading, Random Drops and bought from the TF2 Store
• Vintage - Most commonly, these are items that were Unique, but created before Valve released the 'Mann-conomy' update and made crafting easier.
• Genuine - Items obtained through cross-game promotions (i.e. purchasing another game whilst it offers TF2 items as a bonus). These are mostly cosmetics but also some weapons.
• Strange - Tracks Kills/points. See above.
• Unusual - Items that have unique particle effects when worn. (Until now, mostly cosmetics and new also weapons)
• Haunted - Items released for Halloween updates, which can (generally) only be used/seen on Halloween/Full Moons)
• Collector's - Weapons that were crafted after collecting 200 unique versions of that weapon (along with the 'Chemistry Set' item that creates the 'collector's' weapon)
• Decorated - Uniquely skinned items with rarity-based colours, obtained from completing contracts introduced in the Gunmettle Update (or from unlocking them from Weapons cases)
• Community - Hand-given by Valve to users who have contributed to the TF2 Community - these weapons have a unique particle effect.
• Self-Made - Given by Valve to the weapon creators who submitted the weapon idea on the Steam Workshop
• Valve - You're playing with/against a Valve employee. Ask about Half Life 3 :P.

Weapon level is utterly meaningless in TF2. Every weapon has a set level that cannot be altered or changed in any way, and which ultimately means nothing. It grants no increased damage or other effects of any kind. However, you can gain weapons that have different statistics and abilities through random drops or trading. These weapons all have different characteristics that differ from the default weapons and may suit your playstyle more. An important thing to note is that no weapon is overpowered or more applicable to different situations; all weapons are well balanced and equally effective. It's about finding the weapon that suits you.