Why doesn't this make my TextLabel visible?

The StarterGui is not what the player sees. It is simply how the server knows what to put in each player's PlayerGui.

You're gonna have to do this:

local player = game:GetService("Players"):GetPlayerFromCharacter(h.Parent)
player.PlayerGui.Message.Introduction.Intro.Visible = true

Please use a code block next time.

Regular Scripts cannot access the PlayerGui. To achieve what you are aiming for, you can use RemoteEvents.

First, lets set everything up.

Place a local script inside of the intro frame

Regular Script

local replicatedstorage = game:GetService("ReplicatedStorage")
local event = replicatedstorage:FindFirstChild("ServerGuiAccess")
local plrservice =  game:GetService("Players")

if not event then
    event = Instance.new("RemoteEvent")
    event.Name = "ServerGuiAccess"
    event.Parent = replicatedstorage
end

local debounce = false

script.Parent.Touched:Connect(function(hit)

    if debounce == true then return end

    local plr = plrservice:GetPlayerFromCharacter(hit.Parent)
    if not plr then return end

    local humanoid = plr.Character:FindFirstChildOfClass("Humanoid")
    if not humanoid then return end

    debounce = true

    local tele = game.Workspace:FindFirstChild("Teleporter1part2")
    plr.Character:MoveTo(tele.Position) --shorter way of teleporting character

    event:FireClient(plr) -- sends a signal to the client

    wait(1)

    debounce = false

end)

Local Script

local replicatedstorage = game:GetService("ReplicatedStorage")
local event = replicatedstorage:WaitForChild("ServerGuiAccess")

event.OnClientEvent:Connect(function()
    script.Parent.Visible = true
end)

I have just written this outside of studio, so there may be errors.