How to make a ScreenGui element show upon seating on a seat?

The problem with your script is on the OnClientEvent. Look at this line:

guiEvent.OnClientEvent:Connect(function(plr, seat)

There is something wrong. It's the plr parameter. This is a common mistake with RemoteEvents and RemoteFunctions. Many users may know that the first parameter to OnServerEvent and OnServerInvoke is always the Player that fired/invoked the remote, or because the first argument to Invoke/FireClient is always the Player object to fire/invoke to. They pass it on to OnClient, but is not necessary, since the client doesn't need to know who fired the remote, because the client does NOT fire OnClientEvent. The server is firing the remote to the client.

guiEvent.OnClientEvent:Connect(function(seat)

-- Fixed!

Now another problem with your script is in your Touched event.

seat.Touched:Connect(function(plr)
    cgpEvent:FireClient(plr, 1)
end)

You're trying to fire the remote to a BasePart. You'll have to get the player on your own, the Touched event will not do this for you.

seat.Touched:Connect(function(part)
    local plr = game:GetService("Players"):GetPlayerFromCharacter(part.Parent)

    if plr then
        cgpEvent:FireClient(plr, 1)
    end
end)

Here's the final product:

Server code:

local ReplicatedStorage = game:GetService("ReplicatedStorage")
local cgpEvent = ReplicatedStorage:WaitForChild("ChangeGUIPosition")
local garage1 = game.Workspace.Garages.Garage1
local seat1 = garage1.LaptopTable1.Seat.Seat

seat.Touched:Connect(function(part)
    local plr = game:GetService("Players"):GetPlayerFromCharacter(part.Parent)

    if plr then
        cgpEvent:FireClient(plr, 1)
    end
end)

Client code:

local ReplicatedStorage = game:GetService("ReplicatedStorage")
local guiEvent = ReplicatedStorage:WaitForChild("ChangeGUIPosition")
local seatTouched = false
local owns

owns = script.Parent.Parent.Parent.Value -- The Value is called 'Value'

guiEvent.OnClientEvent:Connect(function(seat)
    if seat == 1 and owns.Value == 1 then
        playerGui = script.Parent
        seatTouched = true
        playerGui.Hint:TweenPosition(UDim2.new(0.5, 0, 0.8, 0))
        playerGui.Hint2:TweenPosition(UDim2.new(0.5, 0, 0.85, 0))
        if seatTouched then -- no need for == true!
            wait(5)
            seatTouched = false
        end
        if not seatTouched then -- No need for == false
            playerGui.Hint:TweenPosition(UDim2.new(0.5, 0, 1.1, 0))
            playerGui.Hint2:TweenPosition(UDim2.new(0.5, 0, 1.15, 0))
        end
    end
end)

Ok, I was playing with the script for a while and was discussing it with @userdetail on discord and now it works. Here's how the scripts look now:

Script:

local cgpEvent = game.ReplicatedStorage.ChangeGUIPosition
local garage1 = game.Workspace.Garages.Garage1
local seat1 = garage1.LaptopTable1.Seat.Seat

-- Changed the whole function so it doesn't happen too often thanks to @userdetail
seat1.ChildAdded:connect(function(child, plr)
    if child:IsA("Weld") == true then
        cgpEvent:FireClient(plr, 1)
    end
end)

LocalScript:

local guiEvent = game.ReplicatedStorage.ChangeGUIPosition
local seatTouched = false
local owns

if script.Parent.Parent.Name ~= "StarterGui" then
    owns = script.Parent.Parent.Parent:WaitForChild("Value") -- I've put a :WaitForChild() here
end

guiEvent.OnClientEvent:connect(function(seat) -- I deleted the 'plr' here
    if seat == 1 and owns.Value == 1 then
        playerGui = script.Parent
        seatTouched = true
        playerGui.Hint:TweenPosition(UDim2.new(0.5, 0, 0.8, 0))
        playerGui.Hint2:TweenPosition(UDim2.new(0.5, 0, 0.85, 0))
        if seatTouched == true then
            wait(5)
            seatTouched = false
        end
        if seatTouched == false then
            playerGui.Hint:TweenPosition(UDim2.new(0.5, 0, 1.1, 0))
            playerGui.Hint2:TweenPosition(UDim2.new(0.5, 0, 1.15, 0))
        end
    end
end)

EDIT: This script works only on Studio unfortunately.