Katra said:

They look a lot like bean pods so I can't help wondering if they are
edible. ;-)

I suspect they are; the leaves are edible, too, but pretty tough, and the
stalks are edible once you peel the tough outer layer away.

Radish pods are used in Asian cooking. Maybe broccoli pods could also
be used. (Both are in the same family.) The broccoli pods could be too
tough or stringy, though. Try nibbling one raw to test for tenderness.

Do I need to let these die back and dry naturally, or can I cut the
stalks or remove the pods to let them dry?

If you want to let the seeds mature so you can turn them into
nutritious broccoli sprouts, you have to let the pods dry on the
plant.

How do I know when the seeds are mature???

The pods will be dry and the seeds will be dark.

This is my first time EVER growing brocolli, and I'd really like to yank
these plants to put in new ones! I need the space......

Well, then maybe you don't want to wait for the seeds to mature.
Harvest the pods (if they aren't too tough and stringy to eat) and clear
the space for the next crop.

--
Pat in Plymouth MI ('someplace.net' is comcast)

Any technology distinguishable from magic is insufficiently advanced.
(attributed to Don Marti)