For the specific case of Delta, there are a few things going on here (and, for what it's worth, I frequently fly Delta and have Platinum Medallion status with them.)
First of all, this is not the upgrade list that another answer mentions, nor does it usually have anything to do with overbooking or irregular operations (equipment change, cancellations, etc.) There is indeed a waitlist for upgrades to Comfort+ or First Class, but it's a separate list from the "awaiting seat assignments"/standby list. Upgrade waitlisted passengers typically do have a seat assignment, but are just hoping for a better one and will not appear on the standby/awaiting seat assignment list.
There are multiple categories of people who may be awaiting seat assignments at the gate on a Delta flight:
Frequent Flyers on Same-Day Standby
One of the most common categories of people on the "Awaiting Seat Assignment"/Standby list are frequent flyers using Same-Day Standby.
People who fly Delta frequently enough to earn Gold, Platinum, or Diamond Medallion frequent flyer status have same-day confirmed and same-day standby fees waived. With this benefit, a business traveler who, for example, finishes a meeting early may choose to join the same-day standby list for an earlier flight than the one they are booked on, as long as it's on the same day. This allows them to get home sooner if they finished their business earlier than anticipated. This benefit is used very frequently by business travelers and will account for quite a lot of the people on the standby list on most Delta flights.
Anyone can do Same-Day Standby (or Same-Day Confirmed) flight changes, but it's mostly used by frequent flyers who get it for free. It costs $75 if you don't have at least Gold Medallion status. Frequent flyers also have a higher priority on the standby waitlist.
Passengers booked in Basic Economy
Passengers booked in Basic Economy class are confirmed on a flight, but may not receive their seat assignment until they get to the gate on heavily-booked flights. They generally will get a seat, but might not have a particular seat assignment until shortly before boarding.
Another common category of people traveling on Standby who will appear on this list are non-revenue passengers, that is, airline employees and/or their friends and family. Most airlines allow their employees to standby for empty seats on just about any of their flights for little or no cost, so, as you can imagine, this benefit is used frequently. They will (at least in theory) clear after all of the passengers booked on normal tickets and after frequent flyers on Same-Day Standby.
This is not as common as any of the categories listed above, but overbooking is another way for passengers to end up on this list. However, they definitely do not overbook by 15 passengers. More than likely, your flight was not overbooked at all and this was not a factor.
Irregular Operations (IRROPS)
IRROPS are when flights are cancelled, downgauged, delayed, etc. due to unforeseen circumstances, such as weather, air traffic delays, maintenance problems, crew out-of-position, etc. In these cases, passengers may need to be rebooked standby on another flight if their flight is cancelled, excessively delayed, or downgauged to a smaller aircraft. Like overbooking, this is a far less common reason than the first three (on Delta, at least,) but it does happen sometimes on all airlines. This was most likely not a factor at all on your particular flight, but it does happen from time to time, especially when there are major weather problems.