[Seany (OP)]

How does this work?

[another_someone]

There are two ways I can imagine it working.

It is by no means likely to be air it inflates with.  Most likely, I would have thought CO₂, but possibly some other gas.

The simpler means would be if there was a small canister of compressed gas (e.g. a soda siphon canister), which has a valve opened by the cord.

The other alternative would be a chemical explosion, similar to that used in air bags in cars (although that particular reaction is very fast, and may be too fast to be comfortably used for life jackets), where two reagents are brought together, and the result of the reaction is massive amounts of gas being given off.