The branches of a tree become progressively thinner and weaker as you proceed further outward from the center of the tree's main trunk. Thus, the further you proceed "out on a [tree] limb" the greater the danger you put yourself in.
It's why a cat who pursues a squirrel to the outer branches of a tree puts itself (and its meal) at great risk: the outer branches are able to support the weight of the squirrel alone but not the combined weight of squirrel-and-cat, and both may fall.
The same would apply, for instance, to a person chasing a cat or an elephant chasing a person (though I've never seen an elephant attempt to climbleopard chasing a tree)monkey.