How Do I Remove a Player From a Table When They Leave the Game?

Create an event listener to detect when a player leaves the game.

game.Players.PlayerRemoving:Connect(function(player)
	local index = table.find(playersAlive, player)
	if (index) then
		table.remove(playersAlive, index)
	end
end)

Make sure you define your playersAlive variable at the highest scope possible; it should look something like this:

local playersAlive = {}

while (true) do
	wait(1)
	-- do round stuff
end

game.Players.PlayerRemoving:Connect(function(player)
	local index = table.find(playersAlive, player)
	if (index) then
		table.remove(playersAlive, index)
	end
end)

If you defined the variable inside of the while loop, the event listener at the bottom of the script wouldn't recognize the variable.