CFrame messing up animation?

Having good names for your objects that are in your game and variables in scripts comes with the benefit of organization and efficiency.

One of the problems I think you may have is that your script is running before things load in. When you identify the variable on Line 2 like script.Parent["Animation"], you should use the bracket method when working with tables. You don't have to but it's usually what you use it for.

Instead use :WaitForChild() or :FindFirstChild(). In our case, since we want the script to wait for our objects we will use :WaitForChild()

--Make sure these lead to the right location. If they don't the server will stay on an infinite wait.
local lookAtPart = script.Parent.Parent.Adornee.Parent:WaitForChild("LookAtMe")
local cookingAnimation =  script.Parent:WaitForChild("Animation")

script.Parent.MouseButton1Click:connect(function() 
    local playerWhoClick = game.Players.LocalPlayer

--I had change the name of lookVector to LookAtPart. We also access its Vector3 pos here.
    playerWhoClick.Character.HumanoidRootPart.CFrame = CFrame.new(Vector3.new(-5.75, 3, 62.7), lookAtPart.Position)

    playerWhoClick.Character.Humanoid.WalkSpeed = 0
    playerWhoClick.Character.Humanoid.JumpPower = 0

One of the things you can fix and try using more are for loops. Most beginner scripters always repeat code that you can condense into a small piece of code.

We're going to use a counting for loop to go up to 3 after waiting a second and playing the animation.

local animationTrack = playerWhoClick.Character.Humanoid:LoadAnimation(cookingAnimation)

--For loops help a lot. Find reference for it on the wiki.
--I also don't know why you want the animation to play multiple times?
--Try using a function that plays it until its finish if this doesn't help.
    for i = 1, 3, 1 do
        animationTrack:Play()
        wait(1)
    end

    print("Played.")

    playerWhoClick.Character.Humanoid.WalkSpeed = 16
    playerWhoClick.Character.Humanoid.JumpPower = 50

end)

Make sure the defined variables lead to the right locations of your objects. Also since we're in a local script, the CFrame-ing of the player might not be seen by other players.