Script to make a gui invisible or visible not working. Why is this happening?

1st. Deprecated Code 2nd. You can't access GUIs from starter gui once you're playing. It's cloned to playergui.

uis = game:GetService("UserInputService")
p = game.Players.LocalPlayer
Frame = p.PlayerGui.ScreenGui.Frame
uis.InputBegan:Connect(function(k, gpe)
if not gpe and k.KeyCode == Enum.KeyCode.E then
if Frame.Visible == false then
Frame.Visible = true
else
Frame.Visible = false
end
end)

You can't acces starter gui like that.

What you can do is put the local script inside of a screen gui and then define the frame like script.Parent:WaitForChild("FrameName")

Other possibility is to have your local script whereever you have it now and access it through PlayerGui like this: player.PlayerGui:WaitForChild("ScreenGui"):WaitForChild("FrameName")

Now I won't rewrite your code for you but that's what you should consider

local Player = game.Players.LocalPlayer
local Mouse = Player:GetMouse()
local CoolFrame = Player.PlayerGui.ScreenGui.Frame

Mouse.KeyDown:Connect(function(Key)
    Key = Key:lower()
    if Key == 'e' then
        if CoolFrame.Visible == false then
            CoolFrame.Visible = true 
        else
            CoolFrame.Visible = false
        end
    end
end)

It's player.playergui NOT startergui, startergui contents just get copied to the playergui.

BTW~~~~do not use Mouse.KeyDown as it is deprecated, use the Context Action Service!

Three things, Mouse.KeyDown is deprecated, use the UserInputService. Also, this needs to be on the Client side in a Local Script. And finally, you are changing the StarterGui which will only take into affect after you reset and it gives you a new gui. You need to use the PlayerGui.

local UserInputService = game:GetService("UserInputService")

local Player = game.Players.LocalPlayer

local CoolFrame = Player:WaitForChild("PlayerGui").ScreenGui.Frame

local function onInputBegan(input,gameProcessed)
    if not gameProcessed then
        if input.UserInputType == Enum.UserInputType.Keyboard then
            local keyPressed = input.KeyCode
            if keyPressed == Enum.KeyCode.E then
                if CoolFrame.Visible == false then
                    CoolFrame.Visible = true 
                else
                    CoolFrame.Visible = false
                end
            end
        end
    end
end

UserInputService.InputBegan:connect(onInputBegan)

:lower() does not exist use string.lower() instead or even better use Enum