In one sentence
Vairagya is dispassion — the releasing of grip on outcomes, possessions, identifications — without becoming cold, indifferent, or withdrawn.
The depth — what it actually means
Vairagya is constantly mistranslated as detachment, which sounds dead. It is not coldness. It is unclenching. The person practising vairagya is fully engaged with life but is not grasping at it. They love but do not cling. They act but do not own. Krishna says (6.35) that the mind is controlled by two things: abhyasa (practice) and vairagya (dispassion). Practice alone becomes performance; dispassion alone becomes laziness. Together they work.
Modern application — how to use this today
Vairagya is the half of practice most people skip. Notice today one thing you are clenching around — a relationship, a result, a possession, an opinion. Loosen the grip just slightly. Not letting go; just opening the fist a little. The Gita's vairagya is built one unclenching at a time.