"I mark’d where on a little promontory it stood" (line 2)
There are lots of good and obvious examples of exploration in this poem, but let’s start somewhere a little different. Keep in mind that the speaker of the poem is exploring when he looks at the spider. We listen to him tell us about a discovery he made. This isn’t a major trip, but it is an important one, and the discovery that he brings home gives the reason for the poem to exist.
"Mark’d how to explore the vacant vast surrounding" (line 3)
The spider is an explorer, and Whitman says as much. When it throws out those filaments, it looks for new territory. This is a metaphor for all kinds of explorations. Think of the ships that explored the world, and connected countries which were previously isolated. The spider sends out threads that are just like those ships. It launches them into the air, to see if they land on something.
"Ceaselessly musing, venturing, throwing" (line 8)
The soul and the spider share a lot of things in this poem. So, it isn’t too surprising to find out that the soul is an explorer, too. When it is "venturing," it’s doing what the spider does, exploring unknown territories and discovering new things.