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"
occur at the end of the poem. In this stanza, the speaker is coming back down to reality after having experienced a period of ecstatic identification with the urn. Earlier in the poem, he describes that and expresses his desire to be one of the figures on the urn, forever young, forever in love, forever in springtime.
As he come "down," the speaker reestablishes the distance between himself and the urn. He is not going to achieve union with the urn. However, thinking more soberly, he says that the beautiful urn "tease[s]" him out of "thought" in the same way as "eternity" (death). What he means is that while he was contemplating the urn, he lost all sense of self and lived in the timelessness of the eternal.
He then tells the urn that it will remain a "friend," a comfort, in moments of sadness. He says in the quote that the urn reveals to him that truth and beauty are the same thing in that both bring us to the same peak of self-forgetting. Truth is eternal and unchanging and so is beauty. Beyond that, there is no point in overthinking the impact of beauty by trying too hard to intellectualize it.
As for who is speaking, this is...
tricky. The speaker wants us to believe it is the urn speaking to him. He says "thou" [you, the urn] sayest [say] and then puts the statement about beauty and truth in quotation marks to emphasize that these are the words of the urn. So on a surface level, yes, the urn is speaking. However, the speaker istelling the urn what it is saying to him.