Why do I have to click it multiple times?

The problem is (I believe) related to your use of waits. Instead of using these waits, I would implement a debounce as it is more reliable:

--0-
local WelcomeGUI = true
local Button     = script.Parent
local GUI = script.Parent.Parent.Frame
local Debounce = true
--0-

function CloseIt() --Start of the function
	if not Debounce then return end -- This will exit the function if debounce is false.
	
	Debounce = false
	
	if WelcomeGUI == true then
		GUI:TweenPosition(UDim2.new(0, 400,0, 200),"Out","Bounce")
		WelcomeGUI = false
	else -- You can simply use an 'else' block here rather than an 'elseif'.
		GUI:TweenPosition(UDim2.new(0, 400,0, -200),"Out","Bounce")
		WelcomeGUI = true
	end
	
	wait(1) -- Tweens by default take 1 second.
	Debounce = true
end

Button.MouseButton1Down:connect(CloseIt)

Alternatively, you can use the override argument, like this:

	if WelcomeGUI == true then
		GUI:TweenPosition(UDim2.new(0, 400,0, 200),"Out","Bounce",1,true)
		WelcomeGUI = false
	else
		GUI:TweenPosition(UDim2.new(0, 400,0, -200),"Out","Bounce",1,true)
		WelcomeGUI = true
	end

This will mean that the player doesn't have to wait for the tween to finish before clicking again.