Here's a short explanation.

None of the unlock weapons are straight upgrades. Think of them as sidegrades. They do things a little different, but are equally good. Now, sometimes certain playstyles make you better with certain weapons over their alternatives (for example, as a Heavy, I'm rarely with a permanent medic buddy, so I generally carry the sandvitch, instead of the shotgun).

The two ways to get these unlock weapons. The first is by completing a certain number of achievements for the relevant character. Look at the milestone achievements to see how many you need. For example, completing 5 Sniper achievements gives you the achievement Milestone 1, and gives you the Huntsman.
Details here: http://tf2wiki.net/wiki/Template:Achievements

The other way is through random drop. This doesn't occur when you kill a player holding a unlock weapon, or when you run over a dropped unlock weapon, or anything like that. How it works is, every so often (something like 20 minutes of playtime), TF2 runs a mathematical formula to determine whether to give you a random item, and what item to make it. This can be an unlock weapon, or much more rarely, a hat. The hats just change a certain class's look a bit, and don't affect anything else. Because hats are much rarer than unlock weapons, you're very lucky if one drops for you.

Now, to equip a weapon, either go to the Character Loadout menu on the main page, or, while in a server, press M. This'll bring up the character you are playing's loadout screen (Press back there to get to the other characters). From here you can click on a weapon slot and change what weapon goes there (either the original or the unlock). Click OK, and then either die and respawn, or change classes to the modified class, or touch a fullhealth/ammo cabinet in a spawn, and...TADA!...your character will be holding their new weapon loadout!

I'll just take a second to reiterate that the unlock weapons are not upgrades, they are different options. Each has a weakness and a strength compared to the original weapons.

Here's the full, completely detailed list of weapon differences: http://tf2wiki.net/wiki/Weapons

I really reccomend that you try out every weapon, fiddle with them as you get them, and try to figure out what you like the best. Feel free to ask us in game what each does if you can't remember or want a different explanation than the number-and-stats-heavy differences in that tf2wiki link I posted up there.

Oh, and don't be suprised if you get duplicates of weapons...that happens alot.

Almost forgot: weapon levels don't mean anything. They're largely there as a joke from Valve poking fun at the people who have complained that TF2 is becoming to close to an RPG. In the past, the levels for each new weapon was a random number between 1 and 100, but they are now fixed (Different for each weapon though). Don't worry about them.

*whew* Hope that helps!