Quite simply, he compares a spider, in the midst of weaving its web, with the narrator's soul. 

"A Noiseless Patient Spider" was included in the 1891-92 Leaves of Grass in a cluster called "Whispers of Heavenly Death." It was written in 1885, seven years before Whitman's death. At the time, Whitman was elderly and his health was failing. Many of the poems in this cluster deal with mortality and, contrary to previous clusters such as "Children of Adam" or "Calamus," deal with a sense of loneliness due to impending death.

The spider is an animate being, but it stands "isolated" and "on a little promontory...to explore the vacant vast surrounding." A promontory is a point of high land that overlooks something vast just beyond it. There is nothing between the spider and the promontory but air. Nevertheless, "[i]t launch'd forth filament, filament, filament, out of itself, / Ever unreeling them, ever tirelessly speeding them." It works tirelessly in isolation.

Likewise, the narrator's soul "[stands]." Like the "vacant vast surrounding" which encircles the spider, the soul is "[s]urrounded, detached, in measureless oceans of space." Instead of weaving, it is "[c]easelessly musing, venturing, throwing, seeking the spheres to connect them..." This work in understanding life and one's being is just as tireless and necessary as the spider's weaving. It is also work that requires quiet and patience. This work continues "[t]ill the bridge you will need be form'd, till the ductile anchor hold, / Till the gossamer thread you fling catch somewhere, O my soul."

The narrator is seeking connection, just as the spider weaves a web to connect the "vacant vast surrounding" to the "promontory," allowing him to move through space while remaining grounded. This is likened to "the bridge" the narrator "will need be form'd...the ductile anchor" that should ground his existence. 

The narrator's capabilities, however, as a mere mortal, are limited and fragile like "the gossamer thread" that the spider flings.