Where are you located? Is the "10" following your screen name the default zone GardenWeb assigns when your zone isn't added?

If you are talking the true tiger lilies (Lilium Tigrinum Lancifolium), orange with dark spots and recurved petals, and the black bulbils are forming and dropping from between the leaf axils and stem ...

...In the cooler zones, they're just dropping naturally so they can be planted where wanted right away. If you try and store them, they'll just shrivel. And every bulbil seems to sprout without any human intervention so they need no special care or attention aside from remembering where you planted them.

Next year, they'll look like a shiny lanceolate blade of grass, the second year they'll have a stem clearly identifiable as a lily. In probably 4 years, they can reach blooming size.

If you do plant them elsewhere, put them barely under the soil, picking a place where they won't get stepped on, pulled up or accidentally dug over. No need to water, rain is sufficient.

I have lots of them and love them for their old fashioned charm - but so do the deer. I have to protect them as best I can with sprays, etc. against that. Deer will eat the growing tips and the buds once they form.

You were right about the default setting of 10.......actually I'm in zone 6 it appears.. I'll attach a pic of the Tiger Lily in the backyard.. Sounds like you described!
So, it appears that growing Tiger Lilys aren't something that happens overnight! As I said, I'm new to growing Lilys so info like this is not unexpected..
What do you think about planting some of the seed in pots to get them started and then transplanting next spring? Is that an option or putting them directly in the ground now a better idea?? I have some concerns about weeds overtaking the selected spot outside, etc..
Thanks for your info, and please respond back at your leisure...

Exactly! We're talking about the same lily.

I've let the seeds both fall and plant themselves naturally and planted in pots for redistribution elsewhere in the gardens. I see you've got enough bulbils to pick off (now is fine, they're ready) to experiment both ways. Although they're very hardy, when I plant some in pots I bury the pot in the ground in an out of the way place - mainly just to keep track of them until they're ready to transplant.

When the bulbils fall naturally around the mother plant, they will eventually form a cluster which makes for a spectacular blooming show.

The only thing problematic with transplanting them in the first growing season is they're fragile and you'll probably end up breaking the bulb away from the new leaf sprout - the little bit of foliage is needed for the bulb to replenish itself for the next season. So, I've just left them potted until the second or third year when they're good and strong enough to handle. Being crowded in even a relatively small pot doesn't hurt them.

If you're really new to lilies - you know that late in the fall when the stalk of the lily pictured above has become yellow/brown and woody you can cut it down at ground level or just slightly above - the yellowing/browning means it has done its job of feeding the bulb for next season's growth.