First of all, the literal meaning of "shuttle". It's something used in weaving - the part of the loom that holds the thread and which goes back and forth across the cloth. See
here for a diagram.
So "shuttle" = something that goes to one place, then comes back, then goes again, and comes back...etc, carrying something else.
Do you remember the
space shuttles, like Challenger? They were used to take astronauts up to the International Space Station - they'd go up, come back, go up again, come back (etc), carrying the astronauts.
In aviation, the concept is the same - a plane that goes back and forth between two cities - for example between major cities like New York and Washington there might be one flight an hour. Shuttle flights are mainly targeted at business traveller - who can travel to a different city and get back again with the minimum of time loss.