Why is my Frame not turning visible on a button click?

It's because you're changing it in StarterGui, that won't have an affect. You have to change it through player gui, or through the gui itself.

So, your script should look like this.

script.Parent.MouseButton1Click:Connect(function()
	script.Parent.Parent.SkillsGui.Visible = true
end)

That will alter the visual appearance on the player's screen.

Hope this helped! Feel free to select this as the answer if this helped!