How do I make UI appear when a part is clicked?

As far as I know, listening to the MouseClick event of a ClickDetector should be done from a server script, because the playerWhoClicked is passed in as an argument of the RBXScriptSignal to allow the developer to know which player had clicked the ClickDetector.

Knowing this, the n in your Clicked(n) function definition is the player who clicked. Let's rename it to something more informative, such as playerWhoClicked. Remember, we want this event to be handled in a server script, so you'll have to make some simple changes that would go something like this:

local C1 = script.Parent.Parent.Chassis.ClickDetector

function Clicked(playerWhoClicked)
    local UI = game.ServerStorage.UnitUI:clone()
    UI.Parent = playerWhoClicked.PlayerGui
end

C1.MouseClick:connect(Clicked)

(another minor mistake I found is that you believed that game.Players.LocalPlayer returns a character, when in fact, it returns the actual Player object - but that is unnecessary for this script since it is a server script. Just letting you know for future reference!)