Why did monks and friars take vows of poverty?

Answer 1: Partially it is a form of self discipline, to remove yourself from greed, envy, and covetousness; and to focus your efforts on doing the work of God.

It also ensures that any one member of the monastery or friary community will use a bare minimum of the resources available to it, thereby causing the entire community to consume less, which, in turn, means that there will be more that can be used for such things such as helping the poor.

Answer 2: Friars have always been called to live lives pursuant to the so-called "evangelical counsels," which are vows of poverty, chastity and obedience. With friars, poverty, then, is quite intentional. Many friars are also mendicant, which means that they literally beg for their food, and money to support the friary, and the order.

Starting in the 6th century AD, monks did not vow the evangelical counsels, but did take the Benedictine vows of obedience (to the head of the monastery), stability (committing to a particular monastery), and what was called "conversion of manners" (largely forgoing private ownership of things, and celibate chastity).

It was, in fact, the arrival on the scene of the very mobile (and so stability was, right there, not required) friars, in the 12th century, and their "evangelical counsels" vows of chastity, poverty and obedience, that caused monasticism to start re-thinking their own vows. Not long thereafter, even monasticism began vowing the "evangelical counsels;" and modern Roman Catholicism's canons 654-658 of the Code of Canon Law now require the "evangelical council" or "councils of perfection" vows of chastity, poverty and obedience of all holy orders.

Both monks and friars, then, are now avowed to a life of poverty; they have no choice.

The answer to the question, then, "why did monks and friars live a life of poverty," is that monks, at least, didn't necessarily have to, at first. Friars always did. Eventually monks (and nuns), too, did... and still do...

...because, in the main, they've promised to (though a vow is actually more potent and binding than a mere promise).

If the questioner is asking why the need to require friars and monks (and nuns, and priests, too) to live lives of poverty in the first place, then Answer 1, above, is probably about right.

That answer, in fact, explains why monks starting from the 6th century lived lives of poverty even before the friars came along in the 12th century, with their "evangelical counsels" vows of actual poverty, and caused the church to compel even monks to take vows of poverty, too. Even before monks were avowed to poverty, then, they saw the value of living a life of it so that more could be given to the persons they served... typicallly the poor and impoverished.