Plastisol, the ink generally used for Screen Printing, does not dry by air and must reach an internal temperature near 320-330 degrees F to be fully cured. We have a 20 foot long, 4 foot wide dryer, which is essentially a flat conveyor belt that moves through a long heated chamber, that stays at about 420 degrees. We don't iron or heatpress our shirts after this, as it is completely unnecessary. A regular clothes dryer would not get hot enough to completely cure plastisol ink. Possibly your kit uses some other type of ink?
Here's an article about the Modern Screen Printing Process:
Try the site below for full instructions on how to do silk screening, in four different methods. Or try searching on "silk screening instructions" for additional sites. Good luck!