When you're initially "killed" by an enemy in "Apex Legends," you're not instantly dead. Instead, your character begins a "bleeding out" process. During this process, you can slowly crawl in any direction, and you can continue to spot things. If you're lucky, and your teammate is able to safely do as much, you can be saved from death if someone reaches you before bleeding out ends.
When that timer runs out, a new one begins: If your squadmates are able to grab your banner, they can outright respawn you at a beacon. After that timer is over, though, you're out of the match for good.
Since so many players are used to the permanence of dying in a Battle Royale game, I've seen a lot of players drop out immediately after bleeding out. Don't be one of these people! It often takes a minute to wait for enemies to clear an area before your squadmates are able to grab your banner. There's a strong possibility they'll save you if they can, and you'll know pretty quickly what they're going to do, because you can see their actions live on your own screen.
OF NOTE: If you're downed, and the rest of your squad is still up, try your best to sneak away by crawling to somewhere safe. If you don't have a headset on, you can ping your teammates to your location by spotting on the ground in front of you.