A minimal interface is a style of API design which I contrast here to a HumaneInterface. The idea behind the minimal interface is to design an API that allows the client to do everything they need to do, but boils down the capabilities to the smallest reasonable set of methods that will do the job. (See HumaneInterface for a good example of the difference.)
Using the ruby-array/java-list example from HumaneInterface, the you would not include a first and last method on a list class that already has an indexer and a length method because you can do first and last just using the exising interface. As a result first and last are convenience methods - minimalists don't avoid all convenience methods, but convenience methods have a high bar to cross to get in.
The arguments for HumaneInterface are there, here's the rationale for a minimal interface.
Interfaces take time to learn. A class with a huge interface is not likely to be used well and may well be off-putting in the first place. By keeping a small, focused set of methods, you make it easier for clients to find out what the class is and what it can do.
This focus is also important for the class designer. A common problem with class design is to make classes do too much. Focusing on the essentials helps keep cruft out of the class, allowing it to focus on doing one job and doing it well.
If you follow a humane approach, how do you know where to stop? If you keep adding methods because someone might want them, you'll have no end of methods. So you need some guideline to avoid this explosion of methods. The humane guideline (provide what is useful) is arbitrary and difficult. The minimalist one is simple - if the client can do it with the existing methods, then they don't need an extra one.
(Note that this issue of finding what is useful is more of an issue for people writing published class libraries than it is for application code. With application code you know your uses - it's a closed system.)
If you are using statically typed pure interfaces (such as with the interface keyword in Java and C#), then another reason to keep the method count small is that it reduces the burden on implementers. A large number of methods all have to be implemented which is a lot of work. (Using an abstract class as a mixin can help reduce this burden.)
If you want more functionality on a minimal class you can do it by using other classes. For example in Java if you wish to reverse or sort a list (operations which are regular methods on Ruby's Array) you use the Collections utility class.
When you're working on a library, once you publish it it's very hard to take anything out. As a result it's better to start with something too small and add things, than to have something too big when you can't remove things.