The climax in this poem serves to illustrate how the speaker dwells on the loss of his loved one, Lenore. And some would say he dwells on this to the point of torturing himself. This is a narrative poem, so we can treat it like a story with elements such as exposition, plot, and climax.
In the exposition, the speaker is reading from "a quaint and curious volume of forgotten lore" in order to take his mind off of his grief. He begins to hear a tapping on his chamber door. He investigates it and finds nothing. This "nothing" symbolizes the absence of Lenore. Then the raven flies into his chamber and perches on the bust of Pallas/Athena. The raven represents death and darkness. Positioning itself on top of a symbol of reason and logic, the raven imposes these notions of death and darkness onto logic. The speaker dwells on his sorrow and the death of Lenore; his reason is suppressed.
The plot rises as he continues to question the raven. He is looking for answers, but he knows what the response will be. He wonders if he'll ever be able to forget Lenore. He says "Quaff, of quaff this kind nepenthe, and forget this lost Lenore!" To quaff is to drink and nepenthe induces forgetfulness. He wants to know if he'll ever be cured of his sorrow. He says "Is there--is there a balm in Gilead?" This is a balm that will cure illness. In this case, his illness is his grief. The raven responds "nevermore" to both questions. The climax occurs in the third to last stanza. He directly asks if he will ever see Lenore again, even in heaven (Aidenn). The raven again responds, "Nevermore." This is the climax and it is heightened by the fact that the raven will not leave. Thus, the speaker is faced with never seeing Lenore again and with the raven staying as a reminder, he will never forget his sorrow.