Seedlings grown from seed and nurtured in a controlled indoor environment and to a lesser extent those purchased from nurseries are generally not acclimated to the outdoor environment. They are in a sense 'fragile and tender'.
Hardening off is a simple process of gradually acclimating these tender young plants to the ravages of Mother Nature, in order to ensure their survival in the grown up plant world.
The seedlings are accustomed to a steady temperature, light supply, gentle or very little breeze and basically no environmental stress or disturbances that they will experience in the real world. Temperature extremes, rain, wind and so forth are to be expected in the outdoor realm. Hardening off allows the plants to toughen up and hone their natural defenses.
You can begin the hardening off process indoors by aiming a fan on them, set at a very low setting, a gentle breeze for several hours each day for about a week. After this, place seedlings outdoors in their original containers where they will receive direct sunlight and some wind daily for a week, possibly more. Progressively lengthen the duration they spend outside each day, moving them back indoors at night.
Some gardeners like to put the seedlings in an open box. This is not absolutely essential but is helpful. Moving the plants back and forth, in and out is easier this way.
After a few days, when the seedlings have been left outside for an entire day, move them into full sunlight for a few hours daily gradually increasing this duration as well. After a little more than a week, assuming the calendars last frost date has passed for your region you can plant them in their permanent homes within your garden or landscape.