In the above gif you can see that the face material of a dartboard looks a whole lot like horsehair. And plenty of people will insist that high-quality dartboards are made with horse or camel hair, but it simply isn't true. What's really in a dartboard is called sisal, and it's related to tequila. While sisal is the gold standard today (with cork being used in cheaper models), original dartboards were made entirely of softwoods. This presented a pretty obvious problem: If you keep throwing pointy things at it, eventually the board will become pitted and difficult to use. Pub owners would soak the boards in water to keep lasting damage to a minimum but it didn't solve the underlying issue.
So in the 1930s, bundled fibres of agave sisalana (a relative of blue agave, the stuff used for making tequila) were used on top of a wood backing to reduce damage to the board. Strands of sisal had been used in a similar fashion to hemp for some time, especially in rope-making. They were stiff enough to hold the darts in place, but flexible enough that no hole would be visible once the dart was removed.
As for why animal hair was never used? Some have speculated that animal hair is too smooth or soft, or that camel hair would not have been easily available during the first darts boom in 1930s England. But the simple explanation is that sisal is orders of magnitude cheaper.