BG is Boarding Group and depending on status, card credits or cabin. you might be in BG1 or BG2. Those in BG1 or BG2 and are BE tickets, will be able to have a full size carry on.

Otherwise, BE customers are in BG5. BG5 is the group that can not have full size carry on and will be charged at the gate if they try. Occasionally if the overhead bins become full, BG3 or BG4 (regular economy, no status) will also have to check bags at the gate but will not be charged a fee)

Bottomline, if you are a Basic Economy ticket, and do not have status and do not have a qualifying credit card, you will get the message you got. You will not be able to check-in in advance and you will need to pay for the full-size bag. You will be BG5. As you have a deep discount ticket that was likely noted at purchases as not having free full-size carry-on.

These restricted Basic Economy (available from all the legacy carriers and some LCCs) have been around for a couple of years.