Chewing anything in your mouth, alerts your taste receptors to the 'chemical breakdown' of what is in your mouth. This puts the brain on alert, and the brain sends messages to the rest of the body as to what will need to be absorbed. So, chewing an aspirin
gets the taste buds
to send the message to brain, as to what is coming . . the brain triggers 'responders' throughout your digestive tract that will absorb what is coming faster!
So, it is more than just crushing it and getting it into your system in smaller pieces for faster digestive use. Though that does speed up the process too, because you don't have to wait for it to be broken down further down the digestive tract.
If you were to swallow an aspirin whole with a slug of water, the digestive juices in either the stomach or the lower GI tract would have to break it down 'chemically' into smaller absorbable pieces. By performing 'the mechanical' breakdown' by chewing and mixing the contents with saliva, the process is much faster. Some of the Aspirin can actually be absorbed 'transdermally' through the mouth right away. And the rest is absorbed faster as it travels the digestive tract.
Aspirin is a wonder drug for all kinds of things and the fact that it can minimize a 'heart attack' is really super.
So, why not optimize it's benefits by speeding it along a bit faster into your system, to save your life.