In the context of the poem, "divine despair" refers to a kind of spiritual feeling that seems to be welling up inside the speaker's soul. It is causing him to feel incredibly sad, so much so that he starts to weep.

The speaker is crying, but he doesn't know why. That's why he describes his tears as "idle." All he knows is that the tears come from some "divine despair" that's rising from deep within his heart.

Usually, when we describe something as divine or God-like, it's something good that brings us happiness and joy. But in this case, that's not true. What is divine here is despair, extreme misery at the thought of a lost youth—"the happy Autumn-fields." This misery is compounded at the prospect of old age with all its infirmities.

This aging process is also divine in that it is an intrinsic part of our status as God's creatures. We were all created by God, and we will all one day die. The despair that the speaker experiences, therefore, is ultimately derived from being a creature of God, finite and mortal. There is nothing that he, or anyone of us, can do to change it. No wonder, then, that the speaker is crying so much, and why the autumn fields, rather than filling his heart with joy as they normally would, inspire such "divine despair".