This is an attempt to provide more detail than the previous answers about why a serious cyclist or triathlete might want a power meter. In my view, power-based training can definitely help a lot of riders. However, it requires some cognitive effort to use properly. That effort can be your own or outsourced to a coach. It's definitely not required to progress.
Different cycling events place demands on different energy systems. Century rides, time trials, triathlons (the cycling bit), and similar events demand steady sub-threshold power output. Road races, criteriums, and cyclocross races often demand multiple attacks, usually well above threshold. Targeted, structured training using power is more precise at getting you the sort of stimulus you want. For a long-distance triathlon, you may want mainly tempo and sweet spot intervals. Road racers probably mix VO2max, threshold, sweet spot and possibly anaerobic intervals, and you probably heard advice not to add too much tempo (which means don't take what should be a zone 2 ride and make it zone 3). Criterium racers might lean even more to the anaerobic side. Power measurement also offers a way to assess progress, and a way to measure training load.
An alternative is to just add volume, and if you need intensity, you can often get that in group rides or races. Past a certain point (varies by person), if you just add volume, you're likely to hit a ceiling. But this works for a whole lot of riders!
Another alternative to power is to use perceived effort or heart rate to monitor sessions, and overall fatigue to monitor training load. After all, in the past, cyclists did base miles in the off-season and then they race themselves into shape. As discussed in other answers, this isn't as precise. However, it's definitely possible. For VO2max intervals, you go as hard as you can for 5ish minutes - you don't need to be looking at your power meter for that. It's also possible to get a good sense of what your threshold power feels like, and try to hold that.
As an alternative to measuring progress, you can track some segments in Strava. Yes, this is influenced by environmental conditions, but it's still useful. I'd try to find a 4-5 min segment, usually a hill. Those who live near extended climbs that take 20-60 minutes could use one of those. Or else you can find quieter roads and create a time trial course for yourself.
If you have the choice between either a power meter or a smart trainer, the latter is actually worth considering seriously. Smart trainers make it more feasible to ride through the cold months. Or, while some might think of you as boring, they can actually make structured intervals during the outdoor riding season more feasible, because you don't have to take time to drive or ride out to that hill. For your own rides, you'll need to get a sense of what doing zone 2 (long slow distance) feels like and discipline yourself. Either that or you'll be group riding with others, anyway. In summer, heat management indoors is a problem, and you will need adequate cooling. For example, I often have to use my air conditioner and a fan. Or you could get multiple powerful fans.