There's a quote from the film Heat where Al Pacino says"You can get killed walking your doggie!"

Death is everywhere, it seems that he has fully accepted the fact that all people are mortal and like him, you will die too.

Also, I'm pretty sure that for him it isn't about proving things for other people, it's about how climbing makes him feel about himself.

Different people have different worldviews. Some people consider it not a life well lived (or lived at all) when not going all-in on what it is they do.

There are a lot of passions that are very safe most of the time yet present life threatening risk in peak doses during small moments of time. E.g. crossing an ocean on a sailboat, backcountry skiing, etc

You can die driving to work in the morning. One small mistake on a high speed road and that's it. Alex knows how to climb like you know how to drive.

I'm a skydiver, and before every jump, there's always a chance it'll be my last. Skydiving is an extreme sport that can be practiced safely (but is not, by default, safe).

BASE jumping, while even more dangerous with many outcomes resulting in death, is something I'm considering on trying in the future.

I don't really do it because of the risk. Sure, I love the adrenaline rush and I love doing extreme things, but that's not the core reason. I know many people who have lost a half dozen really good friends each throughout their lives. That is absolutely devastating. Every time there's a fatality, it spreads through the community rather quickly.

I do it because the experience is like no other, being able to fly through the sky (I mean, how can this view[0] not be attractive). It's also an extremely social sport. You get to form close relationships with total strangers rather quickly.

I could also die driving everyday, the odds say I'd have a better chance at dying in a car than skydiving. So there's always the logical route based on historical evidence.

[0]: http://imgur.com/shu3AtY

If you can't find reason in it, doesn't mean others cant.

If you really have to question every action that might end in your death then you would never go outside, put anything in your mouth, go to the bathroom etc. Your question really only makes sense when considering activities that have a relatively high risk of death in which case the answer has to come down to two things. Firstly how risky is this activity? And secondly, how much risk am I willing to take? If you want to answer those questions, I recommend watching Honnold's film on risk as others have pointed out.

Because you die anyway. Just measure risk vs benefit to yourself. Alex climb vast majority of his climbs with rope just like everyone else. Sometimes I guess he feels he wants to do it without, because degree of focus and being in the moment it bring is unparalleled.