I didn't always like puzzle, because as a programmer I need to solve puzzles every day. Guess what - they weren't puzzles I was solving they were problems!
Reason 1: Expressive Outlet
Puzzles are human created problems that can challenge your initiative, skill, ingenuity, intelligence and patience. They are challenges that create a competitive interest whose reward is not fame and glory but self-satisfaction. Generally they are amusements, so they can be good at teaching how to be a good loser! They keep the mind active and can provide an outlet to pent up talents and interests which do not have an opportunity to find expression in your job or every day chores.
Reason 2: Mentally Satisfying
From a psychological point of view, puzzles are highly educational because they direct your thinking along logical channels. They neither ask nor demand physical prowess or mental aptitudes, nor are they limited to any age group. People who are timid can find that puzzles help them to win some degree of recognition which in turns helps to build confidence.
Reason 3: Played Anytime, Anywhere
Unlike physical sports, puzzles are not seasonal (although they can be themed for seasons). There is no strenuous physical activity involved, instead they exercise your neurons. A large playing area is not required, and in some cases no playing area is needed. Many puzzles can be replayed, with each play helping you to build up stronger ways of critical thinking.
Reason 4: Inexpensive
Puzzle games on mobile devices are quite inexpensive or free, but even to make a physical puzzle doesn't need to be expensive or require strong crafting skills.
Reason 5: Easy to Learn
The mechanics of puzzles are quickly acquired. Perfection, exactness and accuracy come with effort expended. Puzzles have a power to create a "pause to refresh" atmosphere because they can played between work tasks, on the bus or any time you need a little distraction.
Puzzled Kitty photo by David K.