usually the teacher would give a stanza of a sutta be reciting it. The student would then try to rebeat it and the teacher would correct him. When it has been grasped, the disciple would seek for a proper place for him, but not to far from his teacher, or elders, so that they would correct him in cases of wrong, and rebeat the stanza. Once gotten, he would go out for the next and after a while he would be able to cite the whole sutta.
Today books and writting down as support or mostly even replacement, would be normal.
My self not really eager and interested in memorizing words and also because the usual way of transfere is very seldom (a proper given) and also not very gifted in that, coul master a "Blessing-Sutta" in two days. (Its quite disturbing when used to meditation, since it appears disturbing like if still carring music lyrics if not abound already)
How ever, the training it self is a really good concentration training and young monks would not do and train it on silent remute places, but in the middle of the "noice" of even hundred other chanting monks. An enviroment which would cause western or modern really problems but their concentration and mindfulness is very good trained to be able to turn off evertthing else.
It's also not seldom that jhanic states are reached and some also become somehow addicted, whould chante the whole night loadly, if their teacher would not restrikt it for a good.
Generally the learn method is total different to modern ways in still traditional countries. Children would learn also in school in a citing way first. Mastered words and remembering, then would be the time rippe to learn the language and other that the meaning while train in precepts.
The benefit is that concentration is trained by an certain object and the mind does not become so full of ideas and speculations, tousands of thoughts.
Modern learning does not really put much into basic mind training good for an easy life but more in making people effective in thinking how to still there desires and be productive outwardly.
Of course in just reciting memorizing texts, mostly without understanding even the language, does not bear much in regard of the path at first glimbs but given that most will not gain fruits or paths, one might leave this world with more ease and not so disturbed and confused like others having developed much ideas. Saññā (Rememberance) is gone anyway with the break up of the body but habits, such as being skilled in remembering, mindfulness and concentration will follow in some ways.
So if one searches for a good practice and finds some tendency to it, it's recommented if lighter objects of concentrations are not so graspable for now.
The more the five hindrences are dominant and people are used to contact and following them, the more difficult will bethis training. Especially when coming in contact with modern live and emense sensuallity.
Used to speak load, one trains also speech, in form of sound, metrics and words, which is also very benefical. Reading loud is also very recommended in regard of reading learning Dhamma and just in modern, wewtern wirld unusual. That is also the reason why most people (since many had ordained as kids) are very skilled in speech that is charming and pleasant.
So it's out of urgency in any case good (if avaliable) to get and understand the meaning, which Upasaka Saptha Visuddhi explained well in his answer.
We should not forget that in folk-buddhism, eg generally, monks are training chanting to have a means for their livelihood and "income", so its at lage, even traditional, a "acter"-traing without which most would not survive, like today modern learning much Dhamma, study it, and then teach people and "entertain" the layity with a other service for livelyhood. Both, how ever, is actually not really good and wrong livelyhood for monks but counts also for lay people, chanter or scholars: this Dhamma is not given to make a livelihood out of it and good right from the beginning, never to speculate to gain material thing, honore... or a punch of reputations on Bu-SE in citing the same, again and again.
Here a gift blessing "Mangala Sutta" (pali with khmer translation after each stanza) of some young former fellow monks as a amr-file attached. (Note that the Khmer Mahanikaya, or long time folk pronanciation is different to all other countries and in Cambodia it was till short a "pure" oral tradition, maybe even the longest "unbroken" lineage till Khmer-rouge time) The Ven. left, , "youngest and smallest, leading the chant, for example, is one, even young, one does good to pay respect like to rare Elders.
- "Vast learning, skill in handicrafts*, well grounded in discipline, and pleasant speech — this is the highest blessing.
Maha-mangala Sutta: Blessings
(Note: This is a gift of Dhamma and not meant for commercial purposes or other wordily gains.)