The Gemara in Eruvin 54a discusses the idea of learning out loud. It begins with a story of a student of R' Eliezer, who would study quiet, but then, after three years, the student would forget all he has learned:
Rabbi Eliezer had a student who would study quietly, and after three years he forgot his studies.
In Proverbs 4:22, it says:
For they are life to those who find them
The Gemara interprets this to mean, based on the word לְמֹצְאֵיהֶ֑ם "to find"
Do not read: “To those who find them [lemotzeihem],” but rather “to those who express them [lemotzi’eihem],” with their mouth.
The whole idea of learning out loud, is that when singing, speaking etc.. you use your body in it. When doing this, you'll retain more than if you study quietly:
if the Torah is ordered in your 248 limbs, i.e., if you exert your entire body in studying it, it will be secure, and if not, it will not be secure.
The Maharsha on this Gemara explains:
ערוכה כל כו' דריש לי' אלימוד תורה דכתיב לעיל מיניה כי ברית עולם שם לי דהיינו התורה שנאמר בה אם לא בריתי יומם וגו' ואמר ערוכה בכל דהיינו בכל רמ"ח אברים שהדיבור בקול רם מביא הרגשה ותנועה לכל האברים וק"ל:
ArtScroll, in their version of Eruvin 54a, mentions in footnote 3:
Therefore, one should always study in a loud voice, because that stimulates movement in all the limbs of one's body, involving them all in the learning process, and that ensures that one's learning will not be forgotten.
The Rambam, in Mishneh Torah, Hilchos Talmud Torah 3:12 says:
[...] And any who makes his voice heard and the time of his study, his learning is sustained in his hand, but the one who reads quietly forgets quickly.
In Yehoshua 1:8, the Torah states:
This book of the Torah shall not depart out of thy mouth;
The Metzudos Dovid, on Yehoshua 1:8 explains this to mean:
Because through use from your mouth, lest you forget what.
See also Peninei Halakhah (footnote):
“This book of the Torah shall not depart from your mouth. You shall meditate thereon day and night.” Although in-depth study of certain matters is possible and perhaps even preferable to do via one’s thoughts, still, at all other times it is necessary to learn by saying the words out loud (Shulchan Aruch HaRav, Hilchot Talmud Torah 2:12). In that way, the learning one accomplishes becomes clearer and the abstract ideas he learned via thought are understood better. Further, someone who learns out loud remembers his learning better (see Eiruvin 54a). However, the Vilna Gaon (Gra) maintains that even though the essence of one’s learning should be done out loud, nonetheless, even thinking the words and ideas of the Torah is part of the mitzvah, as it says (Joshua 1:8), “You shall meditate thereon day and night,” and “meditating” is thinking.
The Peninei Halakhah mentions the opinion of the Kaf HaChayim:
someone who is reading a book usually will also read with his mouth. Nevertheless, although it is good to take into consideration the opinion of the Gra, one must say a few verses aloud after Birkot HaTorah in order to adjoin the blessing to the learning.