Why do spectrum analyzers generate X-Rays
The warning relates to the display, not the instrument itself.

The HP 8565E uses a CRT monitor as its display. CRTs are electron beam devices, and can emit some X-rays if improperly adjusted and/or inadequately shielded. The notice is reassuring you that the hardware limits on brightness, and the leaded glass built into the screen, are sufficient to protect you from exposure.

This sort of notice is normal for devices containing CRTs; You'll probably find similar notices buried in the manuals of old televisions and computer monitors.