I note that you have asked at least two other questions here on eNotes that have to do with Buddhism. Because of this, I assume that this question is about Buddhist monasticism and not Christian monasticism, which is also a topic that is touched on in history classes. I will try to keep my response general enough to apply to more than one kind of monasticism, but I will also make sure that it mentions Buddhist beliefs where applicable.
There are two main reasons to live a life of humble poverty. The first is so that a person can devote their full attention to their spiritual health. When a person lives in the regular world, they are constantly worrying about things other than religion. They have to think about how they are going to make a living. They have to manage relationships with the members of their family. These things can distract a person from focusing on religion. By removing themselves from the world, monastics can make it easier to live a life in which they focus their mind and their attention on the needs of their religion rather than on the needs of mundane, daily life.
The second reason is because humility and poverty are seen as a good thing, particularly in Buddhism. In Buddhist teaching, all life is suffering. This is because we have too many wants. The wants are not necessarily tangible, though many of them are. Buddhists can achieve Nirvana if they are able to give up their wants and allow themselves to lose their attachment to life in our world. The monastic life of humble poverty should help to make this happen. Because they live in poverty, monastics do not have as many physical wants. They do not want to get nicer clothes or more possessions. By living in celibacy, they try to take their focus off the desire for sex and their desire to please their spouse in other ways as well. By living in humility, they try to take away their desire to be important or to be admired. In short, life in humble poverty allows them to get closer to Nirvana because it facilitates letting go of desires.
(Christians do not explicitly share the idea that life is suffering because we want too many things, but Christians, particularly in the time when monasticism was popular, do believe that the love of money is the root of all evil and that pride goeth before a fall. Thus, removing the desire for material possessions and personal importance help a person become more godly. Thus, a life of humble poverty is valuable in this religious tradition as well.)