For starters, these people rarely tend to list the other side of the ledger than comes from this captivity - lots of animals, primates in particular, tend to live considerably longer in captivity than in the wild, for much the same reasons that you and I live longer in modern society than we would in the wilds of Borneo.
Still, let's take the complaint at face value, and ask the question that the bleeding hearts never seem to get around to asking - why don't they put the animals in larger enclosures?
The simplest answer is cost - double the size of the enclosures and you'll need roughly double the land area to hold the zoo. That means that either the admission cost is going to have to go up, or the zoo will have to be located miles away where land is cheap. Are you willing to pay either of these costs? Probably not.
But I think there's an even more pervasive reason why the enclosures have to be small - humans insist on being able to see the animals close up.
The chimpanzees sure aren't getting any bigger. If you put them in a huge enclosure, then you're more likely to only see them at a distance, or not at all. Not nearly as exciting that way, is it? At a minimum, if you have a really large area, like the wildlife parks or safaris, you need to be able to enter the enclosure to find the animals yourself. It's not hard to see why this model doesn't scale very well if you want to have thousands of people passing through each day, because the potential for accidents becomes enormous. There's a reason they're called "wild animals" - chimpanzees might look cute, but they'll rip your face off.
What people actually want is for the animals to live in a huge natural enclosure, but also to be magically walking by really close at exactly the moment that the person is ready to see them. No such enclosure exists.
Viewed in this light, all the complaints about small cages are just so many crocodile tears, designed to assuage the guilty feelings that visitors feel knowing that they're benefiting from the animal's captivity.
As always, don't be surprised when the zoos cater to your revealed preference for small cages rather than your stated preference for large cages. They won't even mind if you complain about the small cages as you demand their existence, to make your conscience feel better. Very few businesses ever do.