Massage is the wrong term--Implant movement exercises help keep the surgical pocket bigger than the implant.
As seen in the replies, some plastic surgeons feel that breast "massage" is not necessary, but most experienced breast surgeons feel that implant movement exercises help to keep the submuscular pocket (created to enclose the implant) larger than the implant, reducing the likelihood (or severity) of capsular contracture.
Capsular contracture is tightening of the scar capsule around the implant, making the breast(s) feel firm. Without implant movement "exercises" or "massage," the scar your body forms around your breast implants is exactly the same size as your implants--if your capsule contracts even a tiny bit (and most scars tend to contract as time goes by, perhaps stretching later as they mature), you get firmness that ranges from slightly firm to "rock-hard." Baker 1 capsules are somewhat firm; Baker 2 capsules are more firm; Baker 3 capsules are quite firm but without visible distortion; and Baker 4 capsules are hard and visibly deformed.
BTW, this answer assumes smooth implants (saline or silicone), since textured implants, like the teardrop "anatomic" or the "gummy bear" implants, are designed to adhere to the tissues and NOT move. This is because shaped implants rely on being in a specific position to look "normal" or "anatomic" and would look silly if upside down, for example!
BREAST IMPLANTS DO NOT HARDEN, EVER.
However, the scar capsule around your implant(s) can thicken or tighten (caused usually by bleeding or bacteria), making your breasts firm or unnaturally-shaped. Careful surgery, in my opinion, is important to avoid bleeding or bacterial contamination that can lead to excessive capsule formation (capsular contracture), and keeping the surgical pocket mechanically open with careful implant movement exercises can reduce both the incidence and severity of capsular contracture, should it occur.
I have my patients begin gentle implant movement exercises every three hours (when they take their muscle relaxant) while awake, simply pressing on the lower pole of the breast to force the implant to slide upwards in the already-created pocket. I have patients go braless until the implants have dropped into position, after which a bra is worn to prevent further drop or bottoming out. I believe implants should be moved in their pocket until complete healing has occurred (several months), and after a few weeks, can be done while showering or through clothing in the restroom.
(My partner does not have his patients do implant movement exercises, which is why our website information does not include my personal recommendations). Neither of us is "right," and neither is "wrong," but we both (as do our colleagues also answering this and similar questions) strive to have our patients obtain the best possible results. BTW, our incisions are completely covered by adherent Tegaderm dressings, which keep any finger or hand bacterial contamination away from the incisions until completely sealed, while allowing complete visibility of the healing incision (all under-the-skin dissolving sutures).