The things are Phalaenopsis orchid roots and their growth is a good sign.

You might have to repot your orchid in the spring. Here is a link to some notes.

American Orchid Society Notes

Look at the picture of the Phalaenopsis in the Inherited Orchid thread below. Lots and lots of roots

Roots.

Jane

Orchids need to dry out between watering so bottom watering will kill the plants roots then the plant. Try submerging the whole pot in water then leaving in the sink to drain. Lots of growers use spagnum moss and I would guess that's what you'll find in that small plastic pot. Easy for them, not so easy for you. Read the link that Arthur sent you and see about repotting after the blooms stop probably spring to summer.

Thanks a lot! I have been watering from the bottom out of habit :) The flowers are drying now. They will probably completely dry in a week or so. I think I should repot after that.

To repot and when to repot depends on the growing conditions you can provide. In other words where you are in the world. I would never repot mine in winter because my growing conditions are not warm enough.