Definition of swing
By "balance" here we mean "balance". A player who is going to "balance his range" is a player who will be concerned about being impossible to read. For example, on the river, a balanced player will want to have very strong bluffs and hands when betting or raising all in. If a player is unbalanced and bluffs too much on the river or does not bluff enough, he will be easy to read and more experienced players will have no trouble playing against him. If he is balanced, he will be impossible to read.
Do you really have to throw away your range?
If you play games at very high limits or if you always play against the same excellent players, yes, it will be important to balance your range to avoid being readable. On the other hand, if you are playing against amateur players who have a lot of weaknesses, you won't need to balance your range; instead, you will need to exploit your opponents' weaknesses. For example, you are on the river in position and it's your turn to play. If you're up against a very conservative player who folds too easily, bluff him often! If you have a good hand, bet small!? Conversely, against a player who calls too much, don't bluff and bet bigger on your good hands only.