Analogies help you to see the similarities between two things, even if they're different in other ways. For example, planes and birds are different in many ways, but they both fly. A bird is an analogy for a plane.
If you need some inspiration while you design something, try to think of other things that share some similarities with your problem.
Analogies don't only inspire solutions to problems, but also help you understand the problem in the first place. If you feel as if you don't know enough about a problem, or don't know what "type" of problem it is, you can look at other problems that your problem is similar to.
Sue has been asked to design a mobile app for an e-commerce company. She's not sure what to put on the home screen of the app, which the user sees first. For inspiration, she can use an analogy: A mobile app for an e-commerce company is similar in some ways to a shop in the physical world.
Sue researches what it's like to walk into a shop. What does a shop put in front of you when you first arrive? How do they let you know what they offer, or give you a sense of what the shop is there for? The solutions that already exist to these problems in physical shops could work, or inspire other solutions, for Sue's app design.