How do you think the police find out the context behind this comment?
Would it involve... investigating it maybe?
Rigorous investigation is your choice of words not mine. If the police were able to find out the situation in which the comment was made without investigating it I'd be very shocked (a psychic police force? But then even that would be a psychic investigation). First of all they'd require access to the facebook accounts to see exactly what was said and what it was in response to. When told that it related to a League of Legends game, assuming no reference was made to it on facebook, they would potentially need to access LoL's chat logs or at least confirm that said match had occurred.
An arrest can reasonably made simply to question the boy in order to investigate the situation, and in the meantime ensure that the threat that was allegedly made could not be put into action.
Also...
If I see a man in a trenchcoat walk past my house, and think he's a murderer, the police would be absolutely right in politely telling me to get a life.
You're almost as bad at making analogies as the guy who was raving about kidney bombs and blood bullets, you should probably stop. If, however, said man (regardless of clothing) walked by mumbling about how he was going to shoot that ***** in the head when he gets home, you'd have grounds to call the police.
It might turn out he was talking about putting his dog down because she had rabies, but you don't know that.