Decreases the likelihood of bad random encounters Possibility of increasing the odds of special encounters Can grant a prompt to choose whether to engage or ignore a random encounter. (Fallout 2) Decreases the negative effects of bad encounters - dehydration and rockfalls. (Fallout)
In Fallout 2, the Outdoorsman skill's job is to decrease the effects of bad random encounters with the possibility of increasing the odds of special encounters. A successful Outdoorsman check will give the choice of whether or not to have an encounter at all. This does not apply for special encounters however, as they force entering the scene.
The icon is most probably a reference to the Looney Tunes, as the Vault Boy is dressed as Elmer Fudd and is hunting a mutated version of Bugs Bunny.
The Fallout scripts contain two other uses for Outdoorsman, however neither are used in the game. The first is a message displayed when discovering a Mutated Fruit. The other is for generating messages about where the Radscorpions came from in random encounters. (Assumedly related to being able to find the Radscorpion cave for the Stop the Radscorpions quest.)