Stream-of-consciousness writing is when the author takes us into the character's head, and relays all of the thoughts just as they occur. As a result, we hear each and every thought-no matter how random or disconnected-that the character thinks. Or, it is when the author themself describes things in a very flowing, haphazard, disjointed way, moving from one thought to the next like a stream, or like a person who is there, contemplating the scene.
In "The Furnished Room" O. Henry starts off by using stream-of-consciousness to describe the evironment; we get very lengthy descriptions of the man's surroundings. It is as if we are an observer, coming on these scenes for the first time, just like the main character is. These descriptions are so varied and long that it reads almost like we are there, viewing the scene with our own two eyes. This way, we can view the rooms that he enters like we are entering them ourselves, and our eyes are flitting over each and every thing in the room.
Then, as he starts searching for signs of his lost love, O. Henry allows us a peek into the thought processes of the man. We are able to read his mind as he desperately searches, as he organizes his thoughts as he goes through the room. We understand fully, through stream-of-consciousness, each and every thought that pops into his mind as he encounters each item. Of the hairpins, "their triumphant lack of identity", of the black hair bow, it is an "impersonal, common ornament, and tells no tales." We understand fully how he analyzes each object, and feel his desire to find a sign of his lost love. As the reader, we step from being an observer, to being inside his mind, and that makes it much more emotional and poignant, as we feel the intensity of his quest.
I hope that those thoughts help a bit; good luck!