The famous last lines of Keats's poem have been the subject of heated critical debate for over two hundred years. As such, we're not going to be able to settle the matter of their meaning here and now.

However, one can nonetheless offer up a possible interpretation, one that has gained a fair amount of critical consensus over the years. The final words of the “Ode” are spoken by the urn. The urn is saying to humankind the only thing that it, along with all other works of art, can say: “Beauty is truth, truth beauty.”

This is because a work of art, though the product of a particular culture and historical epoch, has the remarkable ability to transcend the time and place in which it was produced. Even though the Grecian urn is thousands of years old, it can still speak to Keats as it has done to many, many people over the years. It can only do this because it is an embodiment of timeless truths that will continue to speak to humanity so long as humanity exists upon this earth.

In Platonic terms, Beauty and Truth—note the capitalizations—are not relative to a specific society or historical era. They are ideas, eternal absolutes that find their embodiment in great works of art such as the Grecian urn in the poem. Such absolute ideals are all that we on earth ultimately know. In other words, true knowledge, according to Plato, is only derived from ideas like Beauty and Truth.

The lines

Thou shalt remain, in midst of other woe

Than ours, a friend to man, to whom thou [you] say'st,

"Beauty is truth, truth beauty,—that is all

Ye know on earth, and all ye need to know"

In brief: Art has the power to communicate the truth of human experience. 

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The line in question is somewhat difficult to explicate, but when taken in context of the rest of the poem, Keats seems to be suggesting that the urn presents a set of messages. Taken together these messages can be identified as truth -- or the conclusive notions taken away from images of life that function as a comment on the nature of that life. 

One way to paraphrase the line "Beauty is truth, truth beauty" is to say that art conveys human knowledge and insights better than any other conveyance of meaning (better than science, perhaps, or better than music).

The urn, after all, is depicting human life in various stages and engaged in various tasks. Youth and joy and sacrifice and, thus, religion are all represented. Furthermore, these ideas are presented in such a way as to maintain their mystery and their enigmatic significance. None of the magic of these aspects of life is lost when represented on the urn.

A repeated suggestion in the poem is that by not speaking and by maintaining an allegiance with silence, the urn is capable of articulating both the substance of life and its more mysterious nature.

Thou still unravish'd bride of quietness,

       Thou foster-child of silence and slow time,

Sylvan historian, who canst thus express

       A flowery tale more sweetly than our rhyme

Opinions are divided whether this is Keats' or the urn's comment. Due to the punctuation of the lines, it is conceivable Keats is declaring that beauty is the only time that a subject's true inner nature truly revealed.

The urn speaks these lines to mankind. They address an age old philosophical question: what is truth? The lines mean that rather than seeking the answer to this question in pure reason, we should seek it in beauty: that beauty is the truest thing humanity can experience.