What on earth went wrong here? [SOLVED]

First, you don't need so much white space in your code, let's fix that first:

open = 0
Cstore = script.Parent.CFrame
local player = game.Players.LocalPlayer

function onClick()
    if open == 0 then
        if player.Backpack:FindFirstChild(script.Parent.ItemName.Value) then
    open = 3
        script.Parent.Open_Door:Play()
        for i = 1,16 do
        wait(0.01)
        script.Parent.CFrame = script.Parent.CFrame * CFrame.new(0.25, 0, 0) * CFrame.fromEulerAnglesXYZ(0, math.rad(-5.625), 0)
        end
    open = 1
    elseif open == 1 then
    open = 3
        for i = 1,16 do
        wait(0.01)
        script.Parent.Close_Door:Play()
        script.Parent.CFrame = script.Parent.CFrame * CFrame.new(-0.25, 0, 0.028) * CFrame.fromEulerAnglesXYZ(0, math.rad(5.625), 0)
        end
    script.Parent.CFrame = Cstore
    open = 0
        else
            print("Player doesn't own key.")
        end
        end
end 

script.Parent.ClickDetector.MouseClick:connect(onClick)

Your code is much easier to look at now.

Now, as for your problem, I'm going to assume you're using a LocalScript. A LocalScript will not work outside of a child of the player, therefore your script will not work.

If your script is a Script, you can't reference the LocalPlayer in a server script, which is why the player is nil.

The MouseClick event already returns the player that clicked the part, so all you need to do is reference the player as a parameter in your function, like this:

open = 0
Cstore = script.Parent.CFrame

function onClick(player) --The event gives the player as a parameter, so you can just add a variable to reference that parameter in the parenthesis.
    if player.Backpack:FindFirstChild(script.Parent.ItemName.Value) then
        if open == 0 then
            open = 3
            script.Parent.Open_Door:Play()
            for i = 1,16 do
                wait(0.01)
                script.Parent.CFrame = script.Parent.CFrame * CFrame.new(0.25, 0, 0) * CFrame.fromEulerAnglesXYZ(0, math.rad(-5.625), 0)
            end
            open = 1
        elseif open == 1 then
            open = 3
            for i = 1,16 do
                wait(0.01)
                script.Parent.Close_Door:Play()
                script.Parent.CFrame = script.Parent.CFrame * CFrame.new(-0.25, 0, 0.028) * CFrame.fromEulerAnglesXYZ(0, math.rad(5.625), 0)
            end
            script.Parent.CFrame = Cstore
            open = 0
        end
    else
        print("Player doesn't own key.")
    end
end

script.Parent.ClickDetector.MouseClick:connect(onClick)

See how I rearranged your code so it checks if the player has the key first? This is because the player would not need the key if the open variable was one of the other 2 options.

I hope my answer helped you. If it did, be sure to accept it.

I think I figured out what's wrong. You had a few problems here:

open = 0
Cstore = script.Parent.CFrame

function onClick(player)
    if open == 0 then
        if player.Backpack:FindFirstChild(script.Parent.ItemName.Value) then
            open = 3
            script.Parent.Open_Door:Play()
            for i = 1,16 do
                wait(0.01)
                script.Parent.CFrame = script.Parent.CFrame * CFrame.new(0.25, 0, 0) * CFrame.fromEulerAnglesXYZ(0, math.rad(-5.625), 0)
            end
            open = 1
        elseif open == 1 then
            open = 3
            for i = 1,16 do
                wait(0.01)
                script.Parent.Close_Door:Play()
                script.Parent.CFrame = script.Parent.CFrame * CFrame.new(-0.25, 0, 0.028) * CFrame.fromEulerAnglesXYZ(0, math.rad(5.625), 0)
            end
            script.Parent.CFrame = Cstore
            open = 0
        else
            print("Player doesn't own key.")
        end
    end
end 

script.Parent.ClickDetector.MouseClick:connect(onClick)

You tried to define the player as LocalPlayer, which can't be done unless the script is a LocalScript, and it's inside the Player. To do this with a ClickDetector, define it as an argument in the function. In your script, I just put it as "player", and kept the rest of the script the same.

Also, try to keep your script a bit cleaner looking like it looks now. (It's easier for others to edit and help you this way)

Basically, that was your only problem, but it should be fixed now. Anyways, if you have any further problems, or questions, please leave a comment below. Hope I helped :P