It runs out of memory. The linux kernel kills it. Droplets don't come with any 'swap' by default, so when the memory gets full, it kills a process or won't let you start a new one. Kinda crazy.
Here's an article walking you through adding a swapfile:
https://www.digitalocean.com/community/articles/how-to-add-swap-on-ubuntu-12-04
Note: I still had some problems with compiler processes being 'killed' even after adding a swapfile when 'swappiness' was set to 0. Upping it to 20 fixed the problem.
Note2: You really shouldn't use webrick in production as others have mentioned in the comments. Still, a question is a question :)