How was Kyle Reese sent by John Connor?

The Kyle Reese that is shown during the movie (and that ultimately turns out to be John's father) has been sent through time into the past (our present, when the movie is set) by the adult John from the future. But there is yet another version of Kyle Reese existing as a child (or maybe not even born yet) parallel to the movie's action, who later grows into the version that John then sends back in time.

No matter if it's a circular time loop or different alternative time lines (whose actual physical validity is to be questioned elsewhere, I guess), the main point to take away from the movie's reality is, that the Kyle Reese from the movie, who dies at the end, is not the one that grows into the Kyle Reese that John later sends back in time (as you already realized, he's dead, and also didn't know Sarah personally). It is the one who has already been sent back in time, with his younger version currently existing as a child (or waiting to be born) in parallel somewhere else.

I agree with those other two mind-bending answers -- there are multiple parallel timelines going on. In one timeline, Sarah Connor gives birth to John Connor. John grows up, meets Kyle Reese, and sends him back in time. By sending him back in time, that creates a second, parallel timeline. In that timeline, Kyle Reese is a grown man when John sends him back, and is still a grown man in 1984. He impregnates Sarah, and THEN he dies.

The thing that gets my mind all wrinkled up is that Sarah then gives birth to John, who grows up to send Kyle back to 1984 AGAIN ... and impregnate Sarah AGAIN, and John grows up AGAIN, and sends Kyle back AGAIN ... and Kyle impregnates Sarah a THIRD TIME, etc. If you focus on that storyline, it's a loop. Hard on the brain!

BTW, notice that any way you look at it, Kyle is a GROWN MAN when he dies. And he dies "after" the GROWN UP John Connor sends him back. He also dies BEFORE the baby John Connor is born. But we can assume there's another Kyle, a child, somewhere else in the world at the same time the adult Kyle is impregnating Sarah and then getting killed.

Whew! I need to go watch something simple. Like Frozen.

He did it in 2 ways.

Firstly, by time machine. Once you've been sent back in time, you still exist. From your date of birth all the way through to your date of death.. Even if that date of death was in the past via a time machine.

Secondly he sent him by enticing him. By showing him a photo of Sarah Connor, which is why he confirms to her that he came back for her. I'm guessing that in Kyle's world, the future doesn't offer much time for hanging around with chicks etc what with lots of angry robots trying to terminate you. So when John Connor probably asked a crowded room full of men - who'd like to go back and help his 30 year old mum out- every hand in the room went up. Kyle couldn't believe his luck when he was chosen by his friend. Having to deal with only 1 robot, (as a posed to hundreds daily), plus on top of that having Sarah to hang out with was an amazing job opportunity.