1. The difficulty in these verses is that the text does not specify which sins are being committed. It would be problematic to say that the phrases ולא תעשו את כל המצוות האלה (vs. 22) and את כל אשר צוה ה' אליכם mean that the penalty here is applied only to someone who transgresses all the commandments. Equally problematic would be the alternative suggestion – that כל here means a transgression of any of the laws of the Torah (see NJPS). See Nahm.’s displeasure with iE who (at least according to Nahm.) understands the verse that way.
Rashbam’s reading of these verses conforms to the standard halakhic approach. (So also his commentary to Lev. 6:23, referring to our verses.) However, instead of bringing proof from vs. 22 as the classical sources do (see Rashi, following Hor. 8a), Rashbam cites vs. 23 as proof. These verses must be referring to idolatry, he says, because it is an infraction that leads to transgressing “everything that the LORD has enjoined upon you.” See also Rashbam’s next comment, s.v. מן היום.
Rashbam’s comment here – that the person who worships idols will ultimately end up transgressing all the rules of the Torah inadvertently – resembles Rashi’s comment (following Sifre Numbers 111) that anyone who believes in idols has denied the entire Torah. But the difference is significant. Rashi (and his sources) say that believing in idolatry is tantamount to rejecting the whole Torah. But they also say that our verses are dealing with the inadvertent idolater. Rashbam reformulates the rabbinic statement to make more sense here in the context: an inadvertent idolator will end up breaking all the laws of the Torah inadvertently.

2. In other words, the phrase, “from the day that the LORD gave His commandment and thenceforth,” means all the rules given since the time that God started to give rules. The Decalogue is the first set of laws given by God and the first laws of the Decalogue are about idolatry.
Rashbam’s interpretation looks similar to Rashi’s but there is a difference. Both exegetes are trying to answer the question: what “mistake” is vs. 22 talking about. Both agree that the mistake is idol worship. Rashi explains that since the text refers both to a rule that God “commanded Moses” (at the end of vs. 22) and to a rule that God “spoke to you” (at the beginning of vs. 23), and since the midrash (Mak. 24a and passim) teaches that God spoke the first two commandments of the Decalogue directly to the Israelites, the commandment referred to here must be the law against idolatry. See Hor. 8a and the commentary of pseudo-Rashi there, s.v. איזו היא מצוה. (On the authorship of that commentary, see Grossman, Ḥakhme Ṣorfat, p. 216, note 275.)
Rashbam, like Rashi, believes that the reference here is to the first two commandments of the Decalogue, but not because of the midrash about what the Israelites heard directly from God’s speech. Instead, he explains that the phrase, “from the day that the LORD gave His commandment,” itself means “from the beginning of the Decalogue.”
In many classical rabbinic sources, vs. 23 is seen as expanding the idolater’s culpability and guilt. Rashi says that the verse means that the idolater has also transgressed the words of the prophets. Sifre says it means that the idolater has also transgressed the words of the commandments to the forefathers. Rashbam accepts the idea of expanding the idolator’s guilt, but suggests simply that transgressing the first laws given by God leads to transgressing the later ones too.

3. Note Rashbam’s language here – that the laws of the Decalogue are the first laws given to the Israelites. This theme appears a number of times in Rashbam’s commentary. See similarly commentary to Gen. 26:5 and commentary to Ex. 15:25–26, and note 77 there.