I think there are two separate contexts in which this applies: sport and self-defense.
Sport training tends to focus on the fight over the course of the match. This is true of any striking combat sport (such as Muay Thai), but is even more true in a point-based sport (like Karate), because there is much less risk of an early round knockout. Being the aggressor allows you to see the patterns in your opponent's defenses and exploit them.
Training to be the aggressor means training how to strike in ways that don't leave you vulnerable to counter-attack, or in ways that expect the counter-attack and respond to it. This can include things like using your footwork to move outside your opponent's centerline, using head movement to be less of a target while you're punching, keeping your other hand up to be ready to block any attacks. It also means that the first time your opponent counters, you register that and prepare a defense for it the next time you use your combination.
In Taekwondo sparring, I really liked opening my combos with a crescent kick. People have different ways of protecting their head. I had different follow-ups depending on the read they gave me. If they would block the kick with their hands, then I'd throw a sharply-chambered roundhouse kick (sort of the opposite of a ?-kick, fake a head kick and then roundhouse to the body). If they would lean back, I'd use the crescent kick whiff to chamber a side kick and then nail them in the chest when they straightened back up. Got quite a few knockdowns with that one.
Being the aggressor allowed me to set those up.
Self-defense is a different story. I've always said there are two surprises in self-defense: the initial surprise of the attack, the initial surprise that the defender is fighting back, and after that it's a fight. If you are in a self-defense situation, you have one opportunity to defend yourself before the assailant realizes you're fighting back. This is where I think a lot of the one-step-defense techniques are at their most effective (especially the kind that I learned in Hapkido). In some cases, this may be more of a counter-attack. But in other cases, it may be more of an attack.
After that initial surprise phase of the encounter, it's back to the same rules as any other combat sport.
Grappling is a different story. In grappling, it's very typical that the person who controls the way the fight goes to the ground is the person who controls the fight. If you get a take-down, it's much more likely that you'll land in advantageous position that's difficult for the opponent to recover. If you pull guard, you are more likely to put yourself in a position that you can use to submit your opponent or come on top. It's very good to be the aggressor in a grappling art.