What does sudden pain and pressure in breast implants mean?

Immediately: After surgery it is probably muscle spasm. Very late after surgery (years) it could be ruptured capsule of scar around the implant.

Depends...: You will be much better off seeking consultation with your plastic surgeon; without a good history and physical examination, online consultants can only speculate. Best wishes.

Many possible causes: As stated by dr. Cohen, it may be caused by a variety of different reasons that will vary by timing after surgery. Soon after surgery could mean a hematoma or seroma or muscle spasm. Delayed causes likely relate to capsular contracture or possible inflammation.