I didn't understand the benefit of a power meter until I started riding on
Zwift with my
Wahoo Kickr (smart trainer). Power (typically measured in watts) is the only way to accurately gauge your fitness regardless of environmental factors (wind, hills, etc.)
On Zwift, you start by performing a "functional threshold power test" (FTP test). Basically, you ride as hard as you can for 1/2 hour and your average power output is considered your FTP. This establishes a baseline for the software algorithms to determine how much power you'll need to generate when you do training rides. It also helps you determine which race category you should join. If you're only putting out 2 watts/kg, you'd best avoid joining the "A" category of racers that put out 4-5 w/kg!
If your FTP is 200 watts, when you do a training ride the software will adjust the resistance of the trainer accordingly to hopefully keep you in that range. If you're working on an endurance training ride, the software might require you to ride at 80% of your FTP for 10 minute intervals. The smart trainer will adjust the resistance to ensure that you keep at 80% FTP. If you don't, the software won't give you "points" for completing the training because you dropped below the required power.
So how does this apply to the real world...
For me, I don't race or care much if anyone else is faster than me, but I'd like to maintain a level of fitness and perhaps 'compete against myself'. Using
Strava and a power meter, you can create a "Strava segments" on your favorite hill or sprint section and compare yourself against yourself each time you ride. Because you're using a power meter that feeds data to Strava, you won't have to worry about a head wind one day and not the other skewing your results... it'll be all about the power you put out in that segment.
I personally don't own a power meter yet (other than the one embedded in my Wahoo Kickr). As stated by the OP, I just can't justify the cost as I don't race and it would just be for my own personal knowledge. I'm hoping that someday capitalism will do it's job and drive down the prices of power meters to a reasonable level... then I'll buy.