Leaderstats string value automatically change?

You aren't setting ranking.Value inside of the Changed connection. You put that part after it, so it only gets set once. To solve that, put it inside the Changed event and set the initial value after the event is connected.

datastore = game:GetService("DataStoreService"):GetDataStore("rating")

game.Players.PlayerAdded:connect(function(player)
	local leaderstats = Instance.new("Model", player)
	leaderstats.Name = "leaderstats"
	local rating = Instance.new("IntValue", leaderstats)
	rating.Name = "Rating"
	local ranking = Instance.new("StringValue", leaderstats)
	ranking.Name = "Rank"

	local key = "user_"..player.userId

	rating.Changed:connect(function(val)
		datastore:SetAsync(key, val)
		if (rating.Value <= 1500) then
			ranking.Value = "Rust I"
		elseif (rating.Value > 1500 and rating.Value <= 2000) then
			ranking.Value = "Rust II"
		elseif (rating.Value > 2000 and rating.Value <= 2500) then
			ranking.Value = "Rust III"
		end
	end)
	
	local init=datastore:GetAsync(key)
	if (init ~= 0) then
		rating.Value = init
	else
		rating.Value = 1500
	end
end)