It is a special discount offered only to guests who are currently staying at a Disney resort. They want to entice you to come back for another trip, so they call it a "bounce back". The bounce back discount varies based on time of year. But there are usually two types, a free dining bounce back for a limited period of time in late summer or a room only discount available throughout the year.
When you stay on-site, generally there's a special offer in your room to get you to come, or bounce, back. Needs to be booked during your stay, including the day you check out even after you leave. Expires at midnight that day, so ideally the OP called last night.
I believe the phone number is (407) 939-1923. I don't remember needing a special code. They were able to look up my reservation in the system to verify that we had been staying at a Disney resort that day.
You call the phone number listed on the postcard (or posted above). The reservation center will confirm that you are a current resort guest and then help you book your future vacation.