About 22 years ago...
The starting point for what became JSF, the first snowflake in today's mighty avalanche, was a US-UK agreement to work on a STOVL fighter to replace the Harrier, which was then reaching the end of its development potential. So part 1 of the answer to the first poster is "to replace the Harrier, particularly in its shipborne role."
In the US, STOVL became wedded to CV and CTOL requirements in 1993-96, to save money. This also meant that the jet would be stealthy, which was a non-negotiable US Navy/USAF requirement. The UK (BAE and RR) parlayed their STOVL experience into a large role in the program, teaming with LockMart, who won the competition.
Meanwhile, the 1998 Strategic Defence Review reaffirmed the need for two new aircraft carriers. This was made possible by the existence of JSF, which was then well under way, because at the time nobody was thinking seriously of a cat-trap solution.
The Harrier force also became "jointed", leading to the dumping of the SHAR. This results in an institutional base for a joint JSF force, which could in theory operate off ships or from short runways.
The F-35B has been the default choice for the new carriers (which are still regarded as necessary under UK defense policy) even though they have grown to the point where they could easily operate cat-trap fighters. Nobody ever stopped and examined that question.
So "why JSF"? 1. It's the chosen aircraft for the carriers and as time goes on it's increasingly expensive to change that choice. 2. It provides a true LO capability to the UK. 3. If all goes to plan
it provides lots of lovely lolly to UK industry because the UK buys 85-130 of the jets but gets almost 20 per cent of the program, when you count BAE's EW system.
If all goes to plan of course, which is when we refer you to other threads