To go out (to a bar or someones place) for a short duration of time.
short trippingshort stayshort tripshort stayingshort
JD: Hey, how's it going. Sorry I can't stay, I've got to be awake at 6 o'clock tomorrow.
Regis: There's nothing wrong with "staying short"/"going short"/"short tripping".
You go to a bar with the intent of being there only to be seen, have one drink, see someone that is conveniently available and GTFO (get the f**k out) as soon as possible.