Why should use cases be considered a good format for gathering and documenting functional requirements?
There are a number of reasons actually. No single reason makes them better than all other approaches to capturing functional requirements, but collectively these reasons make a strong case.
First, a use case captures the sequential behaviour that yields a goal. This is a natural way to describe something, more natural than describing its structure or its non-functional characteristics. For example, if I were trying to efficiently describe what an elevator is, I wouldn't begin by saying 'It has a door on each floor, plus two buttons usually placed one above the other.' I'm more likely to say that it is a device that can carry people between the floors of a building. Behavioural descriptions are richer sources of requirements that people in the problem domain can more easily relate to.
Second, a use case documents sequences of interactions. Other more traditional functional requirements formats can bury this sequential aspect, or make it appear of second level importance. For example, a common strategy for documenting requirements, functional or non-functional, was to form numbered lists of requirements statements based on words like 'must', 'shall', 'may' etc. Sequential behaviour was often inferred from the numbering scheme for these single statements of requirement, but the emphasis was on the fact that each numbered statement was a separate single requirement. With a use case, the structure of the requirement is based around goal-driven sequences of interactions, these being natural ways to learn about or describe a subsystem. In this format, it makes very rich behavioural data available to the developers, in a format that they can easily use to construct the code that realises that behaviour.
Third, a use case is by its very nature already nine tenths of the way to becoming user documentation. If I wanted to know how to use a system, I'd like to be told what to input and what to expect to observe as an output.
Fourth, a use case describes the expected sequence of interactions that yield the goal, plus intermediate observable outputs. It also describes the alternative sequences that branch off the primary sequence when unexpected events or conditions occur part way through the use case. This combination of paths through the use case describe behaviour in a format that lends itself readily for translation into test scenarios. Hence use cases are most of the way toward describing functional requirements in terms of the tests that will be used to confirm those requirements were met.
Because of these reasons, use cases have become one of the most widely used formats for capturing and recording external descriptions of system behaviour prior to system design and construction.