고 계시다

is doing (respectful)

The respectful version of saying someone is in the middle of an action: you're using this as your alarm, you're looking for an easy book, you were eating chicken when we came in, are you enjoying yourselves? The plain equivalent is 고 있다; swapping in this form honours the person doing the action, so you use it about guests, elders, customers, and viewers, never about yourself. It is a staple of interview and broadcast speech, which is exactly where these lines come from.

Register: polite and honorific