This is something done by those connecting to the internet on a network which limits one's daily bandwidth usage. It is the practice of not using one's internet connection for a given day until shortly before the daily bandwidth cap reset time, at which point one would start a data transmission and allow it to continue into the next day. This is done in order to complete a data transmission that would normally breach one's daily limit. By running said transmission across two days, one can harness the the combined bandwidth cap of both days.
Eric wanted do download a 4GB game demo, but his university only allowed him 3GB of bandwidth a day. So he started bandwidth fasting this morning. That way he can start the download just before midnight and let it continue past midnight, into early tomorrow morning. He'll use all of his 3GB allotment for today, and 1GB of his 3GB allotment for tomorrow.