Hardening off means gradually acclimating them to outside conditions. If you moved tender young plants from a warm environment immediately to a cool environment, they would go into shock and falter. Instead, you must adjust them to outside weather conditions slowly. For the first three days of hardening off your seed tray, place it outside, sheltered from wind and rain, during the warmest part of the day for two hours. From the fourth to the sixth day, place your seed tray outside for four hours during the warmest part of the day. On the seventh and eighth days, place them outside for a total of six hours a day. You may have to adapt this depending on weather changes; use common sense. This practice should condition the starts enough to harden them off and prepare them for being planted out in containers.
If the current weather conditions are not conducive to planting out in the garden (i.e., too cold, too wet, etc.), instead of hardening off plants straight away, you can transplant (or pot up) the seedlings. Transplant the seedlings into small pots (four-inch pots from the nurseries work great) filled with regular potting soil and keep indoors under a grow light until the weather allows you to harden off the starts and plant them out.