So, indoor bouldering in general is as safe as it can possibly be, your in a controlled environment with thick cushened mats, spealized shoes and chalk that are designed to keep you glued.
Unless your being silly and goofing around, you'll be okay. Dont get me wrong have as much fun as you like and conquer new routes and heights but just show this sport some respect. there's some strict rules in place so you don't injure yourself and others.
let's take a look..
1 - Never walk close to the walls whilst others are climbing. should be pretty much common sense, but youll be surprised. this is so if they fall, they dont fall on top of you.
2 - Never leave keys, drinks and phones etc, apart from them potentially being stolen, they will injure people when falling.
3 - when falling, always fall on your back, never reach out face forwards with your arms and legs as you may end up injurying yourself.