Put simply, active speakers are full range speakers, or subwoofers that have an amplifier module built-in to them. This means you no longer need to carry an external power amplifier around with you. This can greatly reduce the amount of time it takes to set-up and tear-down at gigs.
Active speakers are also sometimes called powered speakers, or self-powered speakers. Whatever you decide to call them, it is all the same at the end. So, here are some of the reasons why you should consider owning active speakers:
First, they make getting in and out of a gig simple. The only drawback is that you must attach a power cable AND a signal cable to each active speaker. If you were going old school, you would just be running one speaker cable to a passive speaker. Even though you are needing to run an extra cable, you make up for it in the fact that you do not need to carry around an external power amplifier, which can be heavy and cumbersome.
The second thing I love about active speakers is that the manufacturer is able to build them so that everything works together in perfect harmony. The manufacturer builds the woofer and compression driver they want for the active speaker, and then builds an amplifier module that produces just the right amount of power for each speaker component. When you have a passive speaker, that manufacturer has no idea what amplifier you are going to use with their speaker. So, you usually get an active speaker whose components are designed to work together perfectly.
The third great thing about active speaker is that you can add as many as you want to a pro audio sound system. Since you just daisy-chain one speaker to the next using microphone cables, you can go on and on, adding as many as you like. Some active speakers even have built-in digital signal processing so that if you spread out your powered speakers far enough away that delay becomes an issue, you can correct for that with the electronics already onboard.
The next thing to love about active speakers is that every manufacturer of active speakers knows that some gigs are going to require subwoofers. So, many speaker companies make active subwoofers that perfectly match their active full range speakers. As far as setting up a pro audio sound system goes, it doesn't get any easier than that. I do suggest that you use a dedicated external electronic crossover, but you can run signal from a subwoofer stright to a full range active speaker if you need to. Many active subwoofers have a dedicated output just for this purpose.
One final note. When active speakers came out over a decade ago, I was concerned that they would not hold up. I thought that the acoustic energy of the speaker components would destroy the amplifier modules, and for a time I was right. That problem is long gone, and active speakers are now every bit as reliable as their passive speaker / external power amplifier partners.