LocalScript not doing what it's supposed to?

Use a LocalScript instead of a script. This makes thing much easier for you. All you need to do is to create a LocalScript inside either in the StarterPack or StarterPlayer>StarterPlayerScripts, which ever one you prefer.

Also I changed your script:

    for i,v in pairs(game.Players:GetChildren()) do

Instead of a for loop. You can do this instead.

local v = game.Players.LocalPlayer -- This makes things much easier! and Much more simple.

while true do 
    if v.leaderstats.Level.Value >= 2 then 
	  script.Parent.Main.Archer.Visible = true
	  script.Parent.Covers.ArcherCover.Visible = false
  	  script.Parent.Frame.Archer.Visible = true
    end
    if v.leaderstats.Level.Value >= 3 then
        script.Parent.Main.Skeleton.Visible = true
        script.Parent.Covers.SkeletonCover.Visible = false
        script.Parent.Frame.Skeleton.Visible = true
    end
    if v.leaderstats.Level.Value >= 4 then
        script.Parent.Main.LavaBaron.Visible = true
        script.Parent.Covers.LavaBaronCover.Visible = false
        script.Parent.Frame.LavaBaron.Visible = true
    end
    if v.leaderstats.Level.Value >= 5 then
        script.Parent.Main.Pikeman.Visible = true
        script.Parent.Covers.PikemanCover.Visible = false
        script.Parent.Frame.Pikeman.Visible = true
        end
    end
wait(1) 
end 

Just copy and paste this code into the LocalScript you created. I hope this worked for you.