Plain English Meaning
ATC is telling you to climb as fast as your aircraft safely can, right now, until you reach the altitude they gave you. Don't take your time about it.
Full Definition
An ATC instruction requiring the pilot to climb at the best rate the aircraft is capable of, without delay, until reaching a specified altitude. The pilot is expected to use maximum practical climb performance and to advise ATC if unable to comply.
Context Anchor
Used during instrument flying when air traffic control needs an aircraft to reach a higher altitude quickly, often for traffic separation or route management.
Derivation
Expedite' comes from the Latin expedire, meaning 'to free from a snare' or 'to make ready.' In ATC use it means 'do this quickly, without delay' — the controller needs the action completed promptly to keep separated.