There's the well-known disaster communications argument. That is, when the phone lines and cell towers go out, you will still have a means of communication. Also, various ham emergency groups are used to pass information about disasters, assist hospitals, provide communication, etc.
Ham radio is a way to talk with people around the world from all walks of life without the need for any infrastructure.
For me, I often talk on ham radio while in the car driving to/from work on one of the local repeaters (the "magic mountain repeater" for those of you in the LA area). To me, it's a bit more engaging than listening to whatever idle morning show is on the radio. And people often give live traffic reports when commuting, etc.
Honestly, it's a hobby that I find fun as something to do. If you're interested in radio in general, it's one of the few hobbies where experimenting in the RF spectrum is encouraged. HAMs found out that HF waves (shortwave) bounce off the atmosphere as opposed to being absorbed or allowed to pass through, for example. Also, you get to have a cool ID code to use online and offline (the state of CA charges a 1-time fee to make it a vanity plate, as opposed to the annual upkeep of most other vanity plates).