This promotion means that the unit, if defending in woodlands or jungles, will be 20% stronger.
This promotion will make the unit 50% stronger if defending in jungle or forest tiles. More importantly, it will allow the unit to move twice as fast in those tiles. For units with 2 or more moves, this stacks with Mobility for even further movement.
This promotion is commonly used to assist choke tactics, using a fast-moving melee unit to pillage and retreat into favourable terrain.
This promotion will mean that the unit with it will be not only 50% better at defending forest and jungle units than an unpromoted unit, but also at attacking them. This is useful to counter another Woodsman unit or a well-defended wooded point.
On top of that, the healing effect is like a free and more potent Medic I promotion - 15% vs 10%, but it also stacks. Woodsman III and Medic I will heal the same as Medic III, (except on tiles adjacent to the healer) and doesn't require a Great General. Of course, these can all stack together for a massive bonus, and make a particularly potent healer.
Finally, 2 first strikes helps reduce the risk of your super-Medic dying if they attack (and it makes it even better at attacking woods)