It's a JS runtime, not specifically servers though? They essentially can bundle Claude Code with this, instead of ever relying on someone installing NodeJS and then running npm install.
Claude will likely be bundled up nicely with Bun in the near future. I could see this being useful to let even a beginner use claude code.
Edit:
Lastly, what I meant originally is that most front-end work happens with tools like Node or Bun. At first I was thinking they could use it to speed up generating / pulling JS projects, but it seems more likely Claude Code and bun will have a separate project where they integrate both and make Claude Code take full advantage of Bun itself, and Bun will focus on tight coupling to ensure Claude Code is optimally running.