Why won't my teleport button script work?

Well first, you have the arguments inverted for the "Teleport" function. The "player" variable should be first, and the place id second. Like this:

function onClicked () 
    local player = game.Players.LocalPlayer
    if player then
        game:GetService("TeleportService"):Teleport(player, 283976037)
    end
end
game.StarterGui.ScreenGui.TextButton.MouseButton1Down:connect(onClicked)

And as TheAlphaStigma pointed out, you're trying to set an event to something in the StarterGui service, which does not automatically update to individual players. You should use the ancestry chain of the script, by using the "Parent" property.

For example:

-- Also make sure this is a local script, since we're getting the LocalPlayer from the Players service.

function onClicked () 
    local player = game.Players.LocalPlayer
    if player then
        game:GetService("TeleportService"):Teleport(player, 283976037)
    end
end

-- Assuming this script is inside the TextButton GUI, this would work.

script.Parent.MouseButton1Down:connect(onClicked)