How do you make a specific player's chat show up in a GUI ?

You can use the Player.Chatted event which fires whenever the player chats.

Chat Script:

-- Normal Script in ServerScriptService
local Players = game:GetService("Players")
local ReplicatedStorage = game:GetService("ReplicatedStorage")

local Remote = RemoteEvent.UIRemote

Players.PlayerAdded:Connect(function(Player) -- when a player joins
    local OnStage = Instance.new("BoolValue", Player)
    OnStage.Name = "OnStage"
    OnStage.Value = false

    Player.Chatted:Connect(function(Message) -- when the player chats
        if OnStage.Value == true then -- if player is on stage
            Remote:FireAllClients(Message)
        end
    end)
end)

Random Player Picker Script:

-- Normal Script inside ServerScriptService or Workspace
local Players = game:GetService("Players")
local StageCFrame = CFrame.new(11.667, 23.652, 95.261)

local bindableEvent = Instance.new("BindableEvent")

function tpRandomToStage()
    task.wait(5) -- task.wait() is slightly faster than wait()

    local allPlayers = Players:GetPlayers() -- basically :GetChildren() but only choose Players
    local randomNumber = Random.new(os.time()):NextInteger(1, #allPlayers)
    local randomPlayer = allPlayers[randomNumber]
    local randomCharacter = randomPlayer.Character
    local randomHumanoid = randomCharacter:FindFirstChildWhichIsA("Humanoid")

    randomPlayer.CharacterAdded:Connect(function() -- when the player respawns
        bindableEvent:Fire()
    end)

    Players.PlayerRemoving:Connect(function(player) -- when a player leaves
        if player == randomPlayer then -- if the player leaving is the same as the player on the stage
            bindableEvent:Fire()
        end
    end)

    -- PivotTo and GetPivot are both CFrame
    randomCharacter:PivotTo(StageCFrame) -- teleports the player
    randomPlayer.OnStage.Value = true -- sets the value to true
    task.wait(45)
    randomPlayer:LoadCharacter() -- respawns the player and teleports back to spawn
    randomPlayer.OnStage.Value = false -- sets the value to false
end

while true do
    local thread = task.spawn(tpRandomToStage) -- runs the function
    bindableEvent.Event:Wait() -- waits until player is done
    coroutine.close(thread) -- stops the function
end

The script for the GUI:

-- LocalScript inside StarterGui or StarterPlayerScripts
local Remote = game.ReplicatedStorage:WaitForChild("UIRemote") 
local player = game.Players.LocalPlayer

Remote.OnClientEvent:Connect(function(chat)
    player:WaitForChild("PlayerGui"):WaitForChild("RappersChat").Inside.Chat.Text = chat
end)