These very unusual looking growths are called scapes. I have read gourmet chefs like to use scapes. Some garlic growers, believe that you should cut these off the plants as soon as they appear to promote good bulb growth. They do produce flowers, which are cool looking. My experience with them, is if I let them keep growing, they got really stiff at the bottom near the garlic bulb and the bulb was not as big as I would have liked. I believe I will cut these off this year, even though they are interesting and fun to look at.
I harvested my garlic around June/July 2004. I hung them up to dry by their tops on a clothes line that I strung on our front porch. I did not braid them. Once the dirt was dry enough to brush off, I cut the tops off and the extra long roots and brushed the dirt off. I stored my bulbs in a basket-like container with a lid.
The flavor of my garlic so was wonderful. It was so amazing to be able to pull it up out of our own garden and use if fresh. We also grow basil and of course made pesto with our fresh ingredients.