It's easy to imagine it as some sort of famine, where there isn't enough food to feed children, or overcrowding so extreme that everyone lives in slums and it's simply impossible to have kids. If it ever reached that point, a population graph would show a sharp fall, not a steady leveling-off.
When I think about previous generations in my country (Mexico) they seem more care-free. They loved someone, they had babies. One single head of household could afford to buy a house. It was a house too, not some tiny apartment!
I look at my life though and it just seems a little more stressful in a bunch of small ways. The squeeze is gentle, not abrupt, but it's there. Living spaces seem smaller, which adds a little stress. Comparing vs old photos, the city just looks more crowded, more full of traffic, which adds a little stress. In their 20s my parents weren't thinking about the environment, but I am, which adds a little stress. I think about the living standard I could give my kids more than my parents seemed to; I just can't shake the thought that my kids need to stand out from the huge crowds of people there's now, which adds a little stress. At the end of the day all these little bits of stress add up and lead to me not wanting kids any time soon.
Perhaps this is what approaching a population cap is meant to feel like. Just stress here and there, leading to less fertility, rather than some hard physical limit.