put (someone) on hold
When you call a company, they sometimes have to "put you on hold". When you're "on hold", you can't hear the other person. Sometimes music will play while you're on hold. After you've waited for a while, the other person will pick up the call again.
In the example above, the speaker says that the company "had" him on hold instead of "put". The word "had" instead of "put" communicates the feeling of being forced to wait.