Asparagus has male and female plants, it is 'dioecious'. The female plants you may have noticed produce small pea sized berries. The berries / seed pods start off green and ripen to a cherry red. The berries almost look like something we'd like to pop in our mouths for a taste - DON"T DO IT ! They are toxic and will probably give you the worst case of the runs you've ever had, trust me, my father grew them when I was a kid, the few I ate earned me a few days off from school - Oy Vey !
Leaving the berries on the plant will also reduce the asparagus plants yield as energy is diverted into seed production instead of the roots and spears. Picking them off as they appear is a good idea.
Some folks like to let some grow to collect and dry for seed saving purposes. This is easily done, when they turn red, dry them on a paper towel indoors, or sun dry them. I wouldn't advise using a dehydrator as you'll probably kill the seeds. When they are completely dry roll them between your fingers to seperate the seeds from the dry pulp. Store them properly and they can be planted next season. You can also plant some in the ground and some will over winter and germinate in the spring as nature designed them do.
Asparagus is generally started from crowns, but starting from seed is workable. Germination is slow (four weeks or more), and weeds are problematic for young seedlings. Young domesticated asparagus can not compete with other plants, in particular weeds. Plants grown from seed indoors can be transplanted -the following spring -to a permanent location.