How would I make this for only the person who touched it?

-- You can't really pause a function. It will continue to run until the code inside of it is done. However you can just remove the function itself, run it in a loop, and detect if a variable is true or not before running the code.
local Player = game.Players.LocalPlayer
local Label = Player.PlayerGui:WaitForChild("ScreenGui").TextLabel
local InGame = 0
local activated = false
local RATE_PER_SECOND = 1

game:GetService("RunService").Stepped:Connect(function(time, step)
	local increment = RATE_PER_SECOND * step
	if activated == true then
		InGame = InGame + increment 
		if InGame < 10 then
			Label.Text = "0:0"..tostring(InGame)
		elseif InGame < 60 then
			Label.Text = "0:"..tostring(InGame)
		elseif InGame >= 60 then   
			local minutes = math.floor(InGame/60)
			local sec = InGame-minutes*60
			if sec < 10  then
				Label.Text = tostring(minutes)..":0"..tostring(sec)
			else
				Label.Text = tostring(minutes)..":"..tostring(sec)
			end
		end
	end
end)


game.Workspace.Activator.Touched:Connect(function()
	activated = true
end)

game.Workspace.Deactivator.Touched:Connect(function()
	activated = false
end)

This is a local script.

Simply try using a local script.

Alright I got it fixed. Read the --'s because one section goes into a server script, then the rest goes inside a local script.

--Inside a server script located anywhere
game.Players.PlayerAdded:Connect(function(player)
	local activated = Instance.new("BoolValue")
	activated.Name = "Activated"
	activated.Parent = player
end)


game.Workspace.Activator.Touched:Connect(function(hit)
	local humanoid = hit.Parent:FindFirstChild("Humanoid")
	if humanoid ~= nil then
		game.Players[humanoid.Parent.Name].Activated.Value = true
	end
end)

game.Workspace.Deactivator.Touched:Connect(function(hit)
	local humanoid = hit.Parent:FindFirstChild("Humanoid")
	if humanoid ~= nil then
		game.Players[humanoid.Parent.Name].Activated.Value = false
	end
end)


-- Inside a local script
local Player = game.Players.LocalPlayer
local SG = Player.PlayerGui:WaitForChild("ScreenGui")
local Label = SG:WaitForChild("TextLabel")
local InGame = 0
local activated = Player:WaitForChild("Activated")
local RATE_PER_SECOND = 1

game:GetService("RunService").RenderStepped:Connect(function(step)
	local increment = RATE_PER_SECOND * step
	if activated.Value == true then
		InGame = InGame + increment 
		if InGame < 10 then
			Label.Text = "0:0"..tostring(InGame)
		elseif InGame < 60 then
			Label.Text = "0:"..tostring(InGame)
		elseif InGame >= 60 then   
			local minutes = math.floor(InGame/60)
			local sec = InGame-minutes*60
			if sec < 10  then
				Label.Text = tostring(minutes)..":0"..tostring(sec)
			else
				Label.Text = tostring(minutes)..":"..tostring(sec)
			end
		end
	end
end)