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.