Analogous Inspiration is about changing your focus, trying to find other products, organizations, industries, services that somehow could relate to your problem, or a part of it. It is never a good idea to close your mind to what you know or what corresponds directly to your research area and this methods helps you "widen your knowledge", discover new perspectives.
To start you need to split your design problem, find what behaviors, activities, actions it is made of. Then you can start thinking about places, products that also deal with these "sub-problems".
A not so old example is about "track cycling": you might have heard about it during the last Olympics in Rio. Americans bikes had their bicycle drive on the left (it had always been on the right), which gave them a gain of speed. Designers got this idea by observing Nascar cars, where the activity is similar: you keep turning left, again and again, on a track that is quite similar and you need to be the fastest.
You need to create a health tool that has to be used really fast? Check how emergency buttons, doors in a plane... are working. Change the context, find an analogous situation and you will probably see the things differently. Nature can also be a good source of inspiration, where similar situations to your problem can actually be found. Check AskNature to find examples of problems solved with the help of nature analogies.