The Radak that is mentioned is from Shmuel II 15:21.
We find in the Talmud (Nedarim 37נ):
אמר רבי יצחק מקרא סופרים ועיטור סופרים וקריין ולא כתיבן וכתיבן ולא קריין הלכה למשה מסיני
R. Yitachak said: The textual reading, as transmitted by the scribes, their stylistic embellishments,
[words] read [in the text] but not written, and words written but omitted in the reading, are all
halachah from Moses at Sinai.
It is not clear exactly what scope of cases this gemara is referring to, but the above Radak neglects to mention it at all. (See also Radbaz vol. 3 §584)
Abarbanel (Yimiyahu intro.) brings the Radak and rejects his explanation on several fronts. He believes that indeed Ezra HaSofer inspected the texts before he compiled the Tanach, but included variant k'ri to supplement many words that were written incorrectly with respect to proper grammar and context, for one of two reasons:
- The word is written incorrectly to deliver a hidden message that the author wished to impart, and Ezra added a k'ri so that listeners would be able to follow everything on its simplest level.
- The word is simply misspelled or ungrammatical or completely missing from the text etc. due to the author's lack of linguistic expertise and grammatical skills, and Ezra added a k'ri to these to "fix" them. (He didn't want to change it completely for fear of it actually being an incosistency of the first type.)
Maharal (Tiferes Yisrael 66) rejects both the above opinions and insists that it was in fact the authors themselves that incorporated the use of double wordings, k'ri and k'siv in their books. They used k'siv for the hidden meanings and deep imparted messages within the text, while the k'ri was used for a basic understanding of the content. Malbim is also of this opinion.
Yehuda Leib ben-Zeev writes that all the above three opinions are true in some cases. That is, there are some cases where they are indeed "Halach L'Moshe MiSinai", as the gemara states, but these can be limited to those which are subsequently listed in the gemara there (for which clearly chazal had a tradition regarding). Other cases were included by the author as alternative readings such as ובעפלים/וּבַטְּחֹרִים and ישגלנה/יִשְׁכָּבֶנָּה of Deut. 28:27,30. Then most of the rest which are clearly grammatical and typographical errors can be attributed to scribal errors.
R' Shmuel Rosenfeld wrote a short book about all matters pertaining to kri and k'siv. He devotes an entire chapter to the reason for the phenomenon, in which he spends many pages listing the faults with Radak's approach. He believes, like the Malbim, that kri/k'siv are incorporated by the author of the book to impart some message, whatever it may be. This was started by Moshe in the Torah and then was used as a technique by all the authors of the later books. [That's what it means "halacha l'moshe misinai"; i.e. the concept of kri/k'siv was "taught" to Moshe at Sinai, and then used by all the later authors of the Tanach.]