Amateur radio, or "ham" radio, is a hobby centring on radio communications. It is incredibly broad, both in the forms of communications and in the range of related activities. Communications methods range from the original short-wave radio, possibly using Morse code, to bouncing signals off the moon or using amateur satellites.
It is a hobby with a vast scope for experimenting such as building your own radios or antennas or experimenting with the latest technology.
This type of question is relevant to many hobbies, e.g. why build a piece of furniture or make an item from leather when it's easier and quicker to simply buy it? Hobbies provide satisfaction through challenging activities. The challenges in amateur radio are wide-ranging. In the simple form, it's making a contact with someone distant using only your equipment, not relying on the telecommunications infrastructure. Amateur radio sometimes comes to the rescue during natural disasters simply because it is independent of that infrastructure. Other challenges are operating with minimum power or at remote locations such as peaks or parks. There are also a range of competitions for those with a competitive streak.
Overall, amateur radio is a broad hobby centred around communications with like minded people.