The lines, “ugly weed, / clinging on cliffs,” is a failed imagistic metaphor. The speaker concocts a perverted dichotomy between himself and his fellows, whom he identifies merely as "them." Leaving those others, "them," unidentified, however, the speaker takes as his task to castigate those who do not agree with his particular brand of philosophy.
The speaker's opening lines identify him immediately as a poetaster as he mixes a metaphor of flower and horse. Those other people, whom the speaker disdains, are like well-kept flowers in a flower pot, but he says they are "harnessed to a pot of dirt." Horses are harnessed, not flowers. His mixed metaphor may bring on a belly-laugh for which the doggerel is not striving.
The first leg, then, of the dichotomy is the flower, and the second is a weed. Thus the speaker is going to try to convince his readers that being a weed is better than being a flower. Thus he claims that he prefers to be a big ugly weed. And he likens that ugly weed, which also lives fastened to dirt just as the flower in a pot does, to an eagle. The absence of logic here is breathtaking: eagles fly, plants do not! It matters not that the plant lives admired in a pot or grows out on the prairie unseen by anyone; neither will ever take wing and fly away as the eagle definitely will.