Offsets (baby plants) are small copies of the adult plant. They are a way for the plant to increase additional to producing seed. They are genetically identical with the mother plant, whereas seed-grown plants are a combination of the seed and pollen parent.
The advantage for the plant of producing an offset are that the plant is protected by the mother as it grows, it is already in the right environment for that species, it has a better chance of attracting a pollinator when it does flower because it is part of a larger clump.
The disadvantages are that it is in competition with its parent for resources, it will be subject to any pests, diseases and disorders that its parent is, it is unable to take advantage of any traits that might have been gained through a combination of genetic material.
The advantage to you as an air plant owner is that you will have a new plant for free, and you will be able to watch it grow.
In many cases air plants will produce an offset after flowering as most plants are monocarpic. Monocarpic means that they produce one flower only during their lifetime. However, it is also possible for them to produce offsets without flowering.
Although the adult plant won't flower again, it won't immediately drop dead. It still has something to contribute as part of a clump in terms of photosynthesis and anchorage. So don't worry if you buy a plant that is flowering or pupping, you will still see quite a bit more life from it.
Not all offsets appear at the bottom of the plant, T. fuchsii produces one in the middle of the plant.
If you grow other bromeliads or see them in gardens/shop you will see that they produce offsets in a similar way: