The phrase is a metaphor; phones don't literally ring off a hook, but it suggests extreme frequency. His popularity soared, and his phone started ringing off the hook with invitations.
Pro Tip 2/3
Hyperbolic Expression
'Ring off the hook' often exaggerates the number of calls to emphasize the high volume. After the lottery win, my phone didn't stop, it was ringing off the hook all day!
Pro Tip 3/3
Potential Stress
It implies that receiving too many calls can be overwhelming and potentially stressful. She could barely cope with the stress; her phone was ringing off the hook with clients.