Quick definition
The portion of an option's price that would be realized if it were exercised immediately. Intrinsic value is what remains at expiration after all extrinsic value has decayed to zero.
The floor of an option's price
Intrinsic value acts as a floor: an American option cannot rationally trade below its intrinsic value, because a trader could exercise and capture the difference. When an option quotes at exactly intrinsic value, all extrinsic has been squeezed out — this most often happens deep in-the-money or on the day of expiration.
Intrinsic vs extrinsic
Total option price = intrinsic value + extrinsic value. Extrinsic value is everything else — time value, volatility premium, dividend and rate adjustments. When traders talk about theta or vega "burning off," they are talking about the extrinsic component; intrinsic value only moves when the underlying moves.
How Treeova uses it
Treeova's cockpit decomposes each option's price into intrinsic and extrinsic components in real time, so traders can see exactly what they are actually paying for. An agent evaluating an exit uses the intrinsic/extrinsic split to distinguish between "the trade already worked" and "the trade is being carried by decay" — very different situations with very different next moves.