The Sifra at the beginning of Vayikra writes that the word ‘leimor’ should be understood as ‘in order to say’, i.e. in order to repeat over to the Jewish People.
The Ramban in Shemos, 6, 10, brings this explanation but points out that it does not work in all cases. He therefore states, “What is correct in my eyes is that the word ‘leimor’ is to indicate clarification of the matter in all places, and God spoke to Moshe in a complete statement and not a doubtful statement, and not hinting any matters.”
HaKesav VeHaKabbalah argues that there are cases where the Ramban’s explanation also doesn’t work such as Lemech calling his son Noach (Bereishis 5, 29) then adding “leimor…” and cites a few other examples such as the beginning of the Aseres Hadibros (Ten Commandments) where the word leimor is used, though it seems superfluous to say ‘with clarity’. In Shemos 20, 1, he therefore takes the Ramban's idea a bit further and says that the word leimor comes to indicate further explanation and details given over. I.e. the word ‘vayedaber’ or ‘vayomer’ would mean to explain the kelalim - the general principles, and ‘leimor’ would mean to explain the peratim - the finer details of the mitzvos. In Vayikra 1, 1, he gives a similar explanation that the word ‘amirah’ in general (and as used in the word leimor) is to indicate greater depth of understanding. He cites the Vilna Gaon who said that ‘dibur’ is the statement of the Torah as it is written before us, while ‘amirah’ is the inner depth of the Torah and its intent and rationale. At his conclusion, he also cites Naftali Hertz Weisel who understood these words to mean the opposite - dibur (vayedaber) is used to indicate the Torah sheb’al peh, whereas amirah (leimor) is used for Torah shebichsav.
Sifra, Vayikra - see “davar acher…” (sorry, you'll have to copy & paste this link) -
Ramban, Shemos 6, 10 - https://www.sefaria.org/Ramban_on_Exodus.6.10.1/en/Sefaria_Community_Translation?lang=bi&with=all&lang2=en
HaKesav VeHaKabbalah, Shemos 20, 1 - https://www.sefaria.org/HaKtav_VeHaKabalah,_Exodus.20.1.1?lang=bi&with=all&lang2=en
HaKesav VeHaKabbalah, Vayikra, 1, 1 - https://www.sefaria.org/HaKtav_VeHaKabalah,_Leviticus.1.1.5?lang=bi&with=all&lang2=en