The flowers will go brown and die, and in their place will be the seeds. Put a paper bag around the flower umbel and give the stem a good shake. If the seeds are fully ripe, they'll just fall off into the bag. You might get a few dead flowers, too, but that's OK. Some seeds will almost certainly fall onto the ground - this is 'self-seeding' and a good thing if you want more dill plants to grow. It's a bad thing if the plant self-seeds all over the place, though!

Here is a link that might be useful: {{gwi:877876}}

Thanks so much for posting that illustration- I will have to look more closely at my plant and make sure there are seeds under those flowers, and not just dead flowers!
: )
Anne