Respawn menu won't close!?

[EDIT2] Your RespawnHandler script should be replaced with a LocalScript Because, MouseButton1Click can ONLY be detected by a LocalScript ! Since you need to use LoadCharacter , prepare a RemoteEvent named anything you like and insert it in ReplicatedStorage. In this case, I'm naming my RemoteEvent "RemoteEvent" which is the default name

This script below is CURRENTLY your script which should be replaced by a LocalScript. So yes, the thing below should be inserted in a localscript

local RespawnButton = script.parent
--game.Players.CharacterAutoLoads = false , you can't put this in a local script so I'm currently commenting it out
local menu = script.Parent 
local Players = game:GetService("Players") 
local player = Players.LocalPlayer 
local RemoteEvent = game.ReplicatedStorage.RemoteEvent --new variable
RespawnButton.MouseButton1Click:Connect(function() 
    if player then 
        menu.Visible = false 
        RemoteEvent:FireServer(game.Players.LocalPlayer,game.Players.LocalPlayer)
    
    end 
end)

In another normal script in ServerScriptService:

local RemoteEvent = game.ReplicatedStorage.RemoteEvent 
RemoteEvent.OnServerEvent:Connect(function(player)
    game.Players.CharacterAutoLoads = false
    player:LoadCharacter()
end)