The quoted section refers to the fact that one grapple attempt uses up only one attack and not a whole Attack Action.
Thus multiple attempts to grapple can be made in a turn if you have Extra Attacks.
This is also supported by Jeremy Crawford, who says:
If you take the Attack action and have multiple attacks, you can replace any of them with a grapple/shove.
(Thanks to Doval, who found this tweet.)
