How do I manipulate the PlayerGui after I preload with ContentProvider?

Your while loop on Line 6 is the problem.

While loops basically pause the script until they're done, so if you're going to make an endless loop, you need to:

A: Move the while loop to the bottom of your script.

B: Put the while loop into a new script.

But to fix your actual problem:

In the while loop that repeats until RequestQueueSize == 0, you need to put the rotation there as well, and the remove the while loop on Line 6, so:

local Menu = script.Parent.StartMenu
local Frame = Menu.Frame
local Pic = Frame.Huh
local ContentProvider = game:GetService("ContentProvider")

local function LoadAssets(AssetList)
    -- Takes an asset list and preloads it. Will not wait for them to load. 

    for _, AssetId in pairs(AssetList) do
        ContentProvider:Preload("http://www.roblox.com/asset/?id=" .. AssetId)
    end

    while ContentProvider.RequestQueueSize > 0 do
        wait(0.1)
	Pic.Rotation = Pic.Rotation + 5
    end
end

LoadAssets({316291477, 328017149, 328371540, 198614198})

wait(0.1)

Pic.Visible = false

Hopefully I addressed your problem correctly :)