The seeds are ripening now, it would seem...

This is what happens when the seeds start to ripen. Watch out for borers that will drill into the seeds and eat them from the inside, and for squirrels, who will just eat them all before you get a chance to.

What does it mean when you say the seeds start to ripen?

The purpose of most flowers is to form seeds. They are the plants reproductive parts. Sunflowers are composites, the center is made up of numerous tiny true flowers, and each one will form a seed, once it is pollinated. those seeds will then grow to maturity and propagate the plant once fully ripe. If you leave the spent flower head on the plant you will see them swell and harden and the head will eventually dry out, which means the seeds are ripe.

There are many varieties of sunflowers, but they all form seeds in pretty much the same way.