What exactly is your question about: (1) how a class can make sure that a.equals(b) is true if and only if a == b, or (2) how using == instead of equals() improves performance?
To answer the second one first (because it's really simple): equals() is a method. A method call and executing all the statements in the method is more work than simply comparing two values, which is what == does. So, obviously, == is a less expensive operation than calling equals(), so if you can use == instead of equals() that improves performance.
For the first question: Note what == does when you use it on variables of non-primitive types: it checks if the two references on both sides of the == refer to the exact same object. If you want to make a class for which == has the same meaning as equals(), you should make it so that it's impossible to make two separate objects of the class which are "equal". Note that for example enums have this property, therefore it's safe to compare enums with ==, you don't need to use equals().