I don't get Remote Events... Help me script this animation to run!?

You don't need remote events for this. Playing animations on your local character will replicate to other players. So just load the animation locally and play it.

Here's what your code should look like

--LocalPlayer's Camera
local player = game.Players.LocalPlayer
local camera = workspace.CurrentCamera
local character = player.Character

--Crouch Animation
local mouse = player:GetMouse()
local Humanoid = player.Character:FindFirstChild('Humanoid')
local AlreadyCrouched = false

local Animation = Instance.new("Animation")
animation.AnimationId = "http://www.roblox.com/asset/?id=4835659821"

local AnimationTrack = Humanoid:LoadAnimation(Animation)

mouse.KeyDown:Connect(function(Key) 
    if Key == "c" and AlreadyCrouched == false then
            Humanoid.CameraOffset = Vector3.new(0, -1.2, -1.5)
            AlreadyCrouched = true
            AnimationTrack:Play()
            print("Crouched")
    elseif Key == "c" and AlreadyCrouched == true then
            Humanoid.CameraOffset = Vector3.new(0, 0.2, -1.5)
            AlreadyCrouched = false
            AnimationTrack:Stop()
            print("unCrouched")
    end
end)

Keep in mind that since the player's character is replaced everytime they die and respawn, and thus a new humanoid is created, you have to use the new humanoid to load the animation. You can simply just place this local script in game.StarterPlayer.StarterCharacterScripts

If you want to learn more about remote events, there are resources online to teach you how they work. Here is the wiki link on remote events https://developer.roblox.com/en-us/articles/Remote-Functions-and-Events

Here is a wiki link on the client-server structure which you have to understand to understand remotes: https://developer.roblox.com/en-us/articles/Roblox-Client-Server-Model