One specific example: interfaces are a good way of specifying a contract that other people’s code must meet.
If I’m writing a library of code, I may write code that is valid for objects that have a certain set of behaviours. The best solution is to specify those behaviours in an interface (no implementation, just a description) and then use references to objects implementing that interface in my library code.
Then any random person can come along, create a class that implements that interface, instantiate an object of that class and pass it to my library code and expect it to work. Note: it is of course possible to strictly implement an interface while ignoring the intention of the interface, so merely implementing an interface is no guarantee that things will work. Stupid always finds a way! 🙂
Another specific example: two teams working on different components that must co-operate. If the two teams sit down on day 1 and agree on a set of interfaces, then they can go their separate ways and implement their components around those interfaces. Team A can build test harnesses that simulate the component from Team B for testing, and vice versa. Parallel development, and fewer bugs.
The key point is that interfaces provide a layer of abstraction so that you can write code that is ignorant of unnecessary details.
The canonical example used in most textbooks is that of sorting routines. You can sort any class of objects so long as you have a way of comparing any two of the objects. You can make any class sortable therefore by implementing the IComparable interface, which forces you to implement a method for comparing two instances. All of the sort routines are written to handle references to IComparable objects, so as soon as you implement IComparable you can use any of those sort routines on collections of objects of your class.