Plain English Meaning
Controllers are telling you to leave your current route and fly straight to a named point starting from exactly where you are right now.
Full Definition
An ATC clearance authorizing an aircraft to proceed directly to a named fix or waypoint from wherever it currently is, rather than continuing on its previously assigned route. The aircraft's present position becomes the starting point of a new direct leg to the specified destination point.
Context Anchor
Seen when using a GPS navigator during instrument flight, especially after air traffic control clears the aircraft direct to a named point.
Derivation
"Present position" means where the aircraft is at this moment, and "direct" means a straight-line course to the named point. The phrase tells the new leg starts here, not at some upcoming fix.