In the first stanza, the speaker comments on the figures on the urn. Since the figures are frozen images, the bride on the urn is "still" (unmoving) and since she has yet to be ravished, she has "still" not consummated her marriage. The images on the urn, depicting a rustic scene, have survived many years; they tell an ancient story. Therefore, the urn is a "Sylvan historian." The figures on the urn are "silent" and being preserved for so long, the urn is like the "foster-child of silence and slow time." The speaker wonders whether the figures are gods or mortals and wonders what their lives were like.
In the second stanza, the speaker sees some figures playing music. The tunes can not be heard but the speaker considers that the unheard music is more perfect in its abstract and immortal form than if it were to be heard by the "sensual ear." The trees, in their immortalized image on the urn, can never be bare. The lover can never kiss his beloved but the speaker tells him not to grieve because although he can never kiss her, she will never fade and he will always be in the process of loving her. "She cannot fade, though thou has not thy bliss, / For ever wilt thou love, and she be fair!"
In the third stanza, the speaker praises the trees which can never lose their leaves. This "nature" on the urn should be happy to never lose leaves or age. The melodist of the unheard songs should likewise be happy in this immortal, albeit frozen state. But the speaker makes a change here and determines that this immortal, unchanging state (outside of time) is not conducive to happiness for those (like himself) who live in time. Thus, to be forever young, the lover always chasing his beloved, is to be "always panting." This leaves the living, "breathing human passion" with a "heart high-sorrowful and cloy'd," (cloy'd meaning "too much of a good thing").
In the fourth stanza, the speaker looks upon another image of a priest attending to some sacrifice. The speaker can not connect to the religious significance. He then considers that the empty town will always be empty. The speaker's tone has become overtly melancholy at this point as he's moved from the immortal beauty of the images to their frozen loneliness.
In the final stanza, the speaker laments the fact that when his generation is gone, the urn will remain. After the speaker's generation has become old or passed, the urn will say:
"Beauty is truth, truth beauty," --that is all
Ye know on earth, and all ye need to know.
One interpretation is that the urn says "Beauty is truth, truth beauty," and "man" (the person looking upon the urn) responds to the urn saying that this is all the urn needs to know because the urn exists in a frozen state where/when its truth is its beauty and vice versa. The "man" notes that this is all the urn needs to know whereas for living beings, truth and beauty are more subjective, complicated, and fleeting.