Purchase rattan strands that match the color and length of the ones in your piece. Soak the new rattan in water while you prepare the piece for repair.
Locate the broken strands of the rattan. Cut through each broken strand at both ends on the underside of the piece and carefully pull them out of the weave. Squeeze a dot of glue onto the remaining cut ends of the rattan still attached to the piece.
Let the glue dry completely. Check the new rattan and make sure that it is flexible and easy to work with. Trim it shorter if necessary. Attach the end of the first strand to one of the remaining stumps on the underside of the piece with glue.
Wait for the glue to set and weave the rest of the strand into the original rattan pattern. Continue attaching strands to the underside of the piece and weaving them in until you've repaired the entire area. Finish the weave on the underside, trim off the excess rattan and glue the end into place.