This is an attempt to provide more detail than the previous answers about why a serious cyclist might want a power meter.
Different cycling events might place demands on different energy systems. Century rides, time trials, triathlons (the cycling bit), and similar events demand steady power output. Road races, criteriums, and cyclocross races often demand multiple efforts at or above one's maximum aerobic capacity. Serious cyclists who want to optimize their capability for a specific event should consider targeted training programs. Targeted training can enable cyclists to make optimal use of their training time, rather than just doing unstructured rides. Targeted training is definitely not necessary to enjoy yourself while cycling, however.
One could use perceived effort or heart rate to target training. After all, to the extent that cyclists had any targeted training in the past, they likely used perceived effort for most of cycling history. However, as discussed in other answers, power is a measure of your body's actual output. Heart rate and perceived exertion are measures of how hard your body is working, akin to the car's tachometer. However, temperature and other environmental changes or illness could have you working much harder to produce the same power output. Moreover, heart rate lags our efforts, and it can be a poor guide for short intervals. Perceived effort is a very rough guide. Speed, or time to complete a segment of known length, is another possible metric, but it involves having segments conducive to the training objective and is influenced by environmental conditions (e.g. wind speed) and traffic. Thus, power is a better guide to targeting one's output level than the other criteria mentioned.
Power, specifically FTP, can also offer an objective assessment of how much one has progressed. Again, best time on a segment of appropriate length could be an alternative measure, but it's influenced by environmental conditions, and it requires a segment of appropriate length.
The obvious downside of structured training using power on your primary bicycle is that you have to buy a power meter and a head unit capable of reading power. These items have come down considerably in cost since the 2000s, but they are objectively expensive. Many serious cyclists may have more than one bicycle. Transferring power meters across bikes takes time and effort, and the meters may not be easily transferrable (e.g. if you have a pedal power meter based on Look Keo road pedals, you may be unwilling to use them in a gravel event and you may be unable to do so in a mountain bike race). Less obviously, one has to commit to a training plan to use power optimally. This requires your own time and effort to track and plan workouts, or it requires an expense to outsource that job to a coach. Most cyclists do not need power to see considerable progress provided they put in the time to ride, and provided they build in adequate recovery time.