When children are direct modeling addition, they are generally using a counting all strategy: they count out the first set, count out the second set, and then count the whole, starting over from 1. Counting on, by contrast, is when you start with one of the addends (which you don't count) and count on the other addend.
Counting on from first vs counting on from higher
Join, Result Unknown problems are particularly well suited to a counting on strategy. The action in a JRU problem is basically a counting on action: there is some amount to begin with, and then you add in some more--so if you count on by the change amount in a JRU problem, you're just adding on the amount of the change. One of the big steps in abstraction is when, in solving a JRU problem, a child moves from a counting on from first strategy to a counting on from highest strategy. For example, in the problem:
Tricia had 4 butterfly stickers. Her friend gave her 7 more butterfly stickers. How many butterfly stickers does Tricia have now?
The action suggests starting with the first number: 4 and counting on 7 more:
She starts with 4...5, 6, 7, 8, 9, 10, 11. She has 11 stickers.
It's a lot more efficient, however, to start with the larger number and count on:
7...8, 9, 10, 11. She has 11 stickers.
That makes the numbers easier, but it no longer follows the order of the word problem. Being able see that you can get the right answer without following the order of the problem is a big step in abstraction, and in understanding addition in a more general way.
What does counting on look like?
The most common way of counting on involves putting up a finger for each number counted after the start number; in this case the sum is the end number. For example, to find 6+3, a child would count:
Some children may keep track of how many they have counted on in other ways, but this is the basic model for counting on.