Back in the days before computers, every time a call would come in, the call taker would manually create a ticket for each call. These were then fed along a conveyer belt to the appropriate dispatcher for them to dispatch when they got them. This took a few minutes for the call taker to complete the card, send it to the dispatcher, have them read it...then dispatch it. When a high priority call came in, the call taker would phone the dispatcher directly. This is where the term "hotshot" came from.
These days, the call taker inputs the call on his/her terminal and it gets sent to the dispatcher. This is still time consuming for the call taker to key in the call and all that jazz. When you call with a priority call, the call taker will get just enough information (where and what) to have units on the way. They'll put the caller on hold, then call the dispatcher directly. As soon as the "hotshot" phone (which is actually just an incoming line on their Norstar phone), the dispatcher keys up on the radio. This patches the dispatcher's voice to the radio and the phone. The faint voice you're hearing is the voice of the call taker that is bleeding into the radio side of the console.
It's really the fastest way to get high-priority calls over the air right away. It's a pretty smart way of doing it... this way, the units in the field hear the call at the exact same time as the dispatcher so they can head to it as soon as humanly possible.
