I'm trying to make a button that opens another gui, but it won't work. What am I doing wrong?

Your miskate is game.StarterGui. I see you dont know what is the difference with PlayerGui and StarterGui. That's easy, StarterGui is the storage who duplique your UIs in the PlayerGui of the client that enter in the game.

So to modify the UI of a client you must access of her PlayerGui.

Use a LocalScript:

local PlayerService = game:GetService('Players') -- This is the folder that you can see on the explorer.

local Player = PlayerService.LocalPlayer -- LocaScript only
local PlayerGui = Player:WaitForChild('PlayerGui')

local exitButton = script.Parent

local missionGui = PlayerGui:WaitForChild('missionGui')

exitButton.MouseButton1Click:Connect(function() 
    missionGui.Enabled = true
end)

Hope this help, to more questions put below in commentaries.

Starergui does not effect players you would need to edit the gui in PlayerGui. If its a localscript it's easy

local player = game.Players.LocalPlayer
player.PlayerGui.missionGui.Enabled = true

In your case

local player = game.Players.LocalPlayer
local exitButton = script.Parent

local missionGui = player.PlayerGui.missionGui

exitButton.MouseButton1Click:Connect(function() 

    missionGui.Enabled = true
end)

MUST BE A LOCAL SCRIPT

local exitButton = script.Parent
local missionGui = exitButton.Parent.Parent:WaitForChild("missionGui")

exitButton.MouseButton1Click:Connect(function()
     missionGui.ScrollingFrame.Visible = true
end)

This is a bit more advanced, but it should work. I wrote it assuming that exitButton is in a gui, not a frame. It works because anything in StarterGui goes into a player's PlayerGui.