Why does Elizabeth believe Wickham so readily, and what does her error reveal about her judgment?

Why she believes him so readily. Wickham arrives in Meryton with an open face, a red coat and a sad story: he says Darcy cheated him of the living Darcy's father meant for him. Austen makes his charm work through Elizabeth's own tastes. She is quick, proud of her ability to read character, and flattered that a gentleman confides in her alone. Wickham tells her exactly what she wants to hear about Darcy, who has just snubbed her at the Meryton assembly and whom she already dislikes. Each new detail he gives confirms the picture she has already formed, so she never has to test it against anything else. Her prejudice against Darcy does the work that Wickham's own honesty never had to.

What the error reveals. Austen has Elizabeth name her mistake herself, late in the book: she realises she was guided by vanity as much as by prejudice, and that she valued the attention Wickham paid her. Her judgment is sharp in company and in small social observations, yet it fails where her pride is engaged. She trusts her quickness of perception so completely that she stops checking the evidence, and she mistakes a persuasive story for a verified one.

The novel's structure supports this reading. Her later reconsideration of Darcy's letter is the other half of the lesson: the same faculty that misread Wickham can correct itself once she is willing to doubt her first impressions.

If you want to test this against the text, look at the scenes where Wickham speaks to Elizabeth in Meryton (Volume I, chapters 15–16) and her reflection in Volume III, chapter 16, where she confesses her own blindness.