Moving weight explosively is more of a mind-set than a specific velocity. You don't need to deload the weight in order to be explosive, rather, you need to be trying to move the weight as fast as possible (on the eccentric portion of the lift). The actual speed of the barbell is unimportant as long as you are trying to rip the weights off the face of the earth.
There are benefits in doing this such as increased muscle recruitment and studies showing increased strength for explosive reps versus controlled reps. You can apply this mindset to your warm-ups, heavy sets, and everything in-between. Additionally, performing reps explosively is simply the easiest method of teaching aggression which translates to higher numbers, IMO.
Since your goal is general strength for health and aesthetics I'd argue that you don't really need to train explosively: slow-and-controlled movements have been shown to promote more muscle growth and a simple progressive overload will still develop strength. However, consider mixing it in! It is fun to put up bigger numbers at the gym. Personally, I train my competition dead-lifts/squats/bench-press explosively, I'll mix in a lot of tempo work on those as well, and for most accessory work I've always liked exploding on the eccentric and then returning very slowly on the concentric.
One thing to be said is, again, performing reps explosively is a mindset. If you're doing an explosive dead-lift it doesn't mean that you jerk your back up and rip your arms out of your sockets to move the bar as fast as possible. It means you're contracting your muscles violently while still using the proper technique. I see a lot of people at the gym that think that if they just raise their back fast they'll get some momentum into the lift -- that's not being explosive, that's just asking for an injury.
There is a difference between moving weights fast and moving weights explosively. Moving weights fast is more for muscular endurance and is usually done in the high rep range.