The Meshech Chochma in last week’s parsha (in his discussion of “lo tasur” here) explains that Hashem foresaw that due to historical or sociological circumstances, additional safeguards would be required for various mitzvos hamitzvos. It was impossible to reveal the what and why of those safeguards in the Torah without also revealing something of those future circumstances, which was impossible, as revealing the future would impinge on our bechira. Therefore, these safeguards were only hinted at but not spelled out.
When the circumstances were ripe for the creation of those safeguards and takanos, Chazal saw in hindsight that the laws they were enacting were already alluded to and present in the Torah itself. These are the asmachtos given for dinim derabbanan. An asmachta is a ratification of Chazal’s thinking as being consistent with the ratzon Hashem.
My son pointed out that this idea is already found in Rishonim (see Ritv”a Rosh Hashana 16), but we both had the same question: according to this approach, how do you read this gemara in Sukkah 6a – link? The gemara there brings a pasuk as being an asmachta to a halacha l’Moshe m’Sinai. A halacha l’Moshe m’Sinai is not a new law enacted at some future point in time that we need justify in hindsight as having always been inherent in the text -- we know halacha l’Moshe m’Sinai was there from day one given at mattan Torah and by definition exists without any basis in text?!