A thimble cage is exactly what it sounds like. It's a cage for holding a thimble, commonly carried (modernly) on necklaces. There are two classes, one that is intended to be purely decorative and another that is decorative but also practical.
The purely decorative types can be used to hold more than thimbles, in jewelry making. Small pretty stones or marbles, even tiny figurines. Or, of course, your decorative thimbles. Though if they're not used to hold thimbles, often they're simply called cages.
The decorative but practical type are intended to be opened without being removed from the necklace so that you can keep your thimble on you and not wherever thimbles go when they disappear. They also work on a chatelaine, which are coming back into fashion for certain crafty types, an older style of carrying useful little bits and bobs on your person, many are beautiful as well as being practical. Many silversmiths who specialize in beautiful thimbles also make thimble cages, partially because they worked really hard on your thimble and don't want you to lose it.
Most people in the sewing arts are familiar with trying to discern what happened to the thimble this time- it seems you lose more than you've ever bought, somehow. They roll in odd and somewhat unpredictable ways, which means it's sometimes hard to figure out what they've rolled under and into. And, of course, if you have cats, the danger is doubled. There is likely a strange hoard of thimbles and hair elastics somewhere in your house that only the cats can get to. The thimble cage is to prevent this. Especially if you've spent a bit of money on the thimble.
You'll commonly find them in silver because of previously mentioned silversmiths. If they specialize in making thimbles, they likely make thimble cages as well, or can direct you to someone who does. People who work that hard at making thimbles don't want you to lose them.
They're generally fairly reasonably priced for silver jewelry, though price really depends on the creator.
You don't have to go with silver, naturally, but it's a reasonable choice. A silver thimble cage is durable and beautiful and the occasional light cleaning will keep it bright. Silver jewelry lasts forever if properly cared for. Also, again, many of the people making thimble cages for practical usage are the same people who make thimbles, and they tend to be silversmiths.