The transporter converts the object at point A to a matter stream then into a pattern buffer (inside the transporter mechanism point B). Only then can it be transmitted again from the buffer to point C. There would be no need to first re-materialize the object at point B.
The matter stream consisted of the stream of sub-atomic particles that resulted from the dematerialization of a transport subject. The stream was relocated from the original site of the subject, passed through the pattern buffer, transferred as an energy beam, and rematerialized into its original form again at its destination (often a transporter platform)°.
The pattern buffer was a key component of transporter systems. The buffer was used to temporarily store the matter stream following dematerialization, but prior to sending the stream to its target. This was done because of the relative motion of transporter and target. By temporarily storing the matter stream, the Doppler compensators had enough time to adjust the targeting scanners¹.
From the Star Trek Encyclopedia:
See the section on Direct Beaming.