He has the right idea with the Touched event on the client, but you must compare the Touched event's parameter with the client to determine if it's actually YOU who touched.
--LocalScript
local plrs = game.Players;
local plr = plrs.LocalPlayer;
local gui = script.Parent;
local part = workspace.part
part.Touched:Connect(function(hit)
local plrThatTouched = plrs:GetPlayerFromCharacter(hit.Parent);
if plrThatTouched == plr then
gui.Visible = true;
end
end)
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