I know you might be thinking to yourself, what is this guy going to tell me about chastity, he is married and has a kid. So obviously he sucks at chastity. Well my answer is that you (person I made up) probably have an incorrect understanding of chastity and what being chaste means. Chaste does not mean not engaging in sexual relations (granted that is one way to think about it), that is more being celibate. Chastity is about not engaging in sexual relations which are contrary to moral or religious values. So I can still be married and be chaste, because I am not engaging in sexual relations which are contrary to my moral or religious values. But I am not going to talk a lot about my personal experiences with chastity, but rather the general concept behind it. Most of this paragraph was pretty much explaining how the Third-Order can hold to this monastic vow.
The monastic concept of chastity is based on Paul’s life, in which he states that if someone can remain unmarried they should, so they can fully devote themselves to God. Without a family or spouse a monastic is better able to devote more time to God. Monks and nuns prove this fact that a single life, lived out in a monastic community, allows one to focus more time on prayer to God. In this way, at least some of the problems of married people and of the world can be lifted from the monk/nun. However, other problems can come from chastity, such as the sin of lust (not to say the sin of lust does not affect non-celibate people).
Many of the early Desert Fathers struggled with lust, the Enemy will try to find any foot hold that he can to tempt those who want to live a life devoted to God. St. Benedict even struggled with lust at one point in his early monastic life. Benedict decided to cure himself of the sin of lust. To do this he threw himself in to a large amount of thorn bushes (naked) and rolled around. He caused himself a lot of physical pain, in order to later remind himself of the consequences of the sin of lust, and to remind himself the pain he felt was nothing compared to the fires of hell. The monastics also have the help of God and their community to help them overcome their sins.
The other purpose of chastity is to keep the monastic with as few worldly ties as possible. In other words if someone does not have to worry about a spouse or child they are more able to move around the world. In some monastic orders, such as the Dominicans or the Franciscans, the monks travel as part of their ministry. There are several stories of Sts. Dominic and Francis traveling to different parts of the world spreading the Gospel. In this way they needed to be detached from certain worldly responsibilities, such as wives, children, and property.