To add to Peppe and PS's points just above, Tim, I think what they are referring to here is that, although there is an "avoid growth" button in the city screen, there may be better options to "stagnate" the city and actually have it be more productive. Here is an example.
Say you have a city that is Size 5 and has reach it's happy cap (5 or 6 happies). This city has some good food resources like corn, pigs or wheat. Also, you have built some mines on hills and also a library. So at size 5 you have 5 citizens working tiles. 3 citizens are working corn, wheat pigs and 2 are working mines. The corn, wheat and pigs are providing a very nice food surplus here. So while clicking on the "avoid growth" button will stop this city from going to size 6 and creating unhappy citizens. What we are saying here is to make your city more productive while the food is less of a concern. So switch one of the citizens working a food resource to working a mine or run a science specialist or two, or work an extra cottage for commerce. Decrease the amount of food worked and increase the amount of hammers, beakers or commerce while this city can't grow anyway.
Another option is to use the whip to prevent growth as this provides production too. Just balance this to the unhappiness you will get temporarily.
Once you find additional happy options like HR, Rep or hook up/trade or happy resources, you can adjust your city accordingly.