Why and how I would use Self-care? (For Self-care users)

The reason to use Self-Care is that it actually saves the team time, as long as you use it appropriately. You spend 32 seconds healing yourself with SC, as opposed to the 16 seconds it takes another survivor to heal you. That's 32 seconds used up overall either way, but by using SC, you save the time it would otherwise take you to locate another survivor and convince them to heal you.

However, if there are healing slowdown perks like Sloppy Butcher in play, using Self-Care is generally a waste of time, unless you're using it in combination with Botany Knowledge, in which case it's usually more efficient than getting another survivor to heal you (the only exception to this is the slowdown from Coulrophobia, or if other slowdowns are stacked, but I won't go into too much detail there unless you want me to). Basically, SC + BK is usually a good idea.

As for when you would use it, the short answer is whenever you would otherwise get another survivor to heal you. It depends on what's going on in the match and what else you have in your loadout. If you're using Adrenaline, for example, and there are only one or two generators left, usually you wouldn't bother healing and just rush the generators instead. If something urgent requires your attention, like a hooked teammate, you should probably avoid healing yourself then as well, unless you need to be able to take a hit in order to rescue them.