It is not just a legend: the goats really climb the trees, even if the plant does not have to be too high and must not have a perfectly vertical and smooth trunk. These animals, in fact, like almost all the members of the sub-family of the Caprinae, have a particular base. Unlike that of the horse, the goat one is made up of two sturdy "nails", under which there is a soft but strong bearing.

(Why do they say "save goat and cabbage"?)

Distributed weight
When the animal lays the paw on the ground, its weight is transferred to the two nails: these widen so as to present a wider area, which therefore gives a firmer grip on the surface where the animal walks or climbs ( there are also goats "climbers"). In addition, the bearing provides an additional grip, which allows the goats to have a safe foot on any surface. The branches of the trees, especially those curved by the wind of the deserts, have a very rough bark, which lends itself well to the "climbing" skills of goats.
(Goats also learn an accent)