Please give me a very basic description of what a stipendiary research fellowship at a university is. Is it just a job that pays you to do research? That, to me, sounds almost too good to be true. What's in it for the employer? Just the prestige? Is there any "catch"?
A stipendiary research fellowship is a research fellowship that has a stipend attached to it. A stipend is similar to a salary. The difference is that a salary is meant to be a payment for selling your labour to an employer (it should be based on the value of your labour), where as a stipend is an allowance to allow you to subsist while you do work for yourself (i.e. it should be based on the cost of living and not the value of the labour). IANAL, but my feeling is that in law and in practice there is actaully very little difference.
The term is frequently used in Oxford and Cambridge University colleges, where the "Stipendary" is used to distinguish those fellows of a collage that are paid an allowance from those whose only reward is to be able to say they are a member of the college - these people will often have employment elsewhere, generally a department of the central university. Note that the stipends attached to a stipendiary research fellowship may not be very large - they are intended either as pocket money for someone that lives and eats in college, or as a nice additional income for someone that has another job on top of their fellowship (probably elsewhere in the university).
For a university "what's in it for the employer" is obvious - they get the research done, which is, after all, the purpose of a university.
Where such a position is offered at a private employer, such a thing might be for two reasons - as you say, it increases the prestige of the employer to say they are a patron of scholarship - part of the companies social responsibility program. But such a person might also serve to bring the latest academic thinking into the company. Companies are often somewhat cut off from the very cutting edge academic research. In theory they could read the papers, but in fact few corporate employees find the time to keep up with the cutting edge. Bringing in some that is up to date, and possibly importantly, has recent and active networks among academics can be a way of keep the company in touch with the latest developments.