How do I get this GUI to replicate to other players' screens?

As DesiredRep commented, you need to use Client Event, which is a part of Replicated Events (and Functions)

You will need the LocalScript you already have, as well as a Server Script within ServerScriptServices For this example, I used a Remote Event inside Replicated Storage named "RemoteEvent"

LocalScript

local player = game.Players.LocalPlayer
local background = script.Parent.Background
local text = background.PlayerName
local debounce = true

script.Parent.Parent = player.PlayerGui
local playername = player.Character:WaitForChild("PlayerName")
local remote = game.ReplicatedStorage.RemoteEvent

remote.OnClientEvent:Connect(function(msg)
    print(msg)
    if msg == "TweenOut" then
        script.Parent.Background:TweenPosition(UDim2.new(0.187, 0, 0, 0), "Out", "Back", 2) 
        text.Text = (playername.Value.." Wins!")
    elseif msg == "TweenIn" then
        script.Parent.Background:TweenPosition(UDim2.new(0.187, 0, -1, 0), "In", "Back", 2) 
    end
end)

while true do
    local stringvalue = player.Character:FindFirstChild("PlayerName")
    if stringvalue ~= nil then
        if debounce == false then return end
        debounce = false
        print("Loop Start")
        function getplayername()
            if player and playername then
            print("Function Start")
            playername:Destroy()
                remote:FireServer("Out")
                wait(5)
                remote:FireServer("In")
                wait(2)
                debounce = true
            end
        end
        getplayername()
    else
        wait(1)
    end
end

Server Script

local remote = game.ReplicatedStorage.RemoteEvent

remote.OnServerEvent:Connect(function(player, setup)
    print(setup)
    if setup == "Out" then
        remote:FireAllClients("TweenOut")
    elseif setup == "In" then
        remote:FireAllClients("TweenIn")
    end
end)

The FireAllClients event can be used to make this occur for all players in the game. The reason you have to send it to the server first is so that it can be then sent to all clients, not just the one player. There is not FireAllClients on the Client-Side.