My here is a very mild oath—an abbreviation of my God with the piece which may give offense left out.
Flies? Birds?, like the Get it? just before, call attention to the very feeble joke the speaker affects to see in the cliché “How time flies ... ”. The story which immediately precedes this closing piece concerns birds and how they fly. The words are cast as questions because the speaker is is giving you hints and in effect asking “Now do you get it?”
LearningEnglish is the name of the BBC series of which this program is an episode, and the speaker announces that more programs in the series will follow shortly after—only seconds after—the current program ends. Away is used after a distance or time measurement to indicate how distant something is in time or space.