Deuteronomy 16:8

On the first day thou shalt offer the omer, and eat unleavened cakes of the old corn; but in the six remaining days you may begin to eat unleavened cakes of the new corn, and on the seventh day you shall assemble with thanksgiving before the Lord your God; no work shall you perform.

Our ancestors have said [Menaḥot 66a] that the six days are to be counted “from when the first standing grain is reaped” [: 9]. It is also plausible that the seventh day is to be added on to the six days, in which case Scripture distinguishes between the beginning and the end of the festival in order to admonish you must not do work. Note that Scripture did not say “any work” [e.g., Exodus 20: 10], because here the meaning is “any servile work” [e.g., Leviticus 23: 7] (in the one place where “any work” is used in this context it is immediately qualified by “except…” [Exodus 12: 16]). Since Scripture has mentioned Passover, during which one is required to visit the Sanctuary, Scripture then proceeds to discuss the second and the third holidays. Thus Scripture does not here mention the New Year, nor Yom Kippur, because on those days the congregation does not gather together in the Sanctuary.

Now he said, And on the seventh day [of Passover] shall be a solemn assembly to the Eternal thy G-d; thou shalt do no work therein. But he did not mention the prohibition of work on the first day [of Passover]; he likewise did not state such a prohibition for the Festival of Weeks, nor for Tabernacles, nor did he mention the commandment of the palm-branch [on the Festival of Tabernacles] and he likewise did not mention here the Day of Remembrance [i.e., the New Year] and the tenth of that month [i.e., the Day of Atonement] — the reason, as I have explained, is that now he came to mention only the [commandment to make an] appearance [in the Sanctuary] and the [commandment of] festivity, for all Israelites are obligated to make a pilgrimage on these three festivals to the place which G-d shall choose and celebrate there every man as he is able, according to the blessing of G-d which He hath given him, to rejoice before Him, to thank for all the good which He hath bestowed on us according to his compassions, and according to the multitude of His mercies.
He [Moses] added, and on the seventh day shall be a solemn assembly meaning: Six days thou shalt eat unleavened bread and in addition to them is a solemn assembly on the seventh day, the purport being that they be further detained before Him to eat the unleavened bread. And because he said that [the seventh day] is Atzereth it might possibly mean that they be “restrained” only for the eating of the unleavened bread as on the [six] preceding days, therefore it was necessary to admonish [also in the same verse] that you are not to do work therein. Similarly He said with reference to the eighth day of the Festival of Tabernacles in the section on the festivals in the Book of Leviticus, Seven days ye shall bring an offering made by fire unto the Eternal; on the eighth day shall be a holy convocation unto you; and ye shall bring an offering made by fire unto the Eternal; it is a solemn assembly; ye shall do no manner of servile work, meaning that [the eighth day] is restrained in addition to the first [seven] days with respect to the bringing of a fire-offering, and it has a further prohibition against [doing therein any manner of] servile work, [a prohibition shared only by the first day of Tabernacles, but not by any of the intermediate days].
And by way of the Truth, [the mystic teachings of the Cabala, the verse stating that the …

וביום השביעי עצרת, on that day the Israelites are to be assembled together in communal worship of the Lord. They will sing the song sung by the Israelites after they had crossed the Sea of Reeds on dry ground. This is why this day has been given a special sanctity.

ולא תעשה מלאכה, if not for that, the seventh day would not be recognisable as holy at all, just as the seventh day of Sukkot is not called a מקרא קודש, holy convocation.