We have commandments of the Joy and Honor of Sabbath. What is the explanation of these mitzvot; what is the meaning and purpose of this commandment?
What do rejoicing and honoring of Shabbat mean, what is it to rejoice and honor a day?
There are indeed two commandments of kavod shabbat (honoring the shabbat) and oneg shabbat (enjoying oneself on shabbat). Kavod shabbat indeed applies to the day of shabbat, but oneg shabbat applies to the Jew: the mitsva is for the Jew to enjoy himself on shabbat.
The Rambam in Mishne Torah Hilkhot Shabbat 30:1 describes these mitzvot:
Kavod shabbat includes preparing the house, to wash himself, to wear nice clothing (different from the week), to prepare a nice meal, etc. This is all in honor of the day, i.e., to make the day special, different, from the rest of the week (similar to kiddush, sanctification, i.e., separating, making special the day).
Oneg shabbat includes eating meat, drinking wine, three meals, not to fast, i.e., making the day a pleasure for man.
There is a lesson in the mitzva of kavod shabbat, in that Shabbat is not only 25 hours in the week, it should illuminate the whole week since preparing shabbat is also a mitzva. It is important enough that the gemara (Kidushin 41a) describes how R Safra and Rava were involving themselves in kavod shabbat, by cutting wood to prepare food or salting fish. They could have had others do it for them, or could have argued they were not required to do it because they were already engaged in learning Torah, but felt kavod shabbat was too important to be left to others.
Shulchan aruch harav 242.1 sums up the Jews view on it very nicely
My try of a partial translation
2 things were clearly said about shabbos by the prophets and they are kovod (honor) and oineg (pleasure) (rambam 30.1) as it says (Isaiah 58:13) "and you shall call the Sabbath a delight, the holy of the Lord honored"
But the root of these is from the Torah (not from the prophets) since shabbos is included (א) in the group of mikrai kodesh [holy callings/ occasions] as it says (vayikra 23.3) "but on the seventh day, it is a complete rest day, a holy occasion..."
Our sages (מכילתא בא פ"ט. סיפרא אמור פ' יב. ספרי פנחס סי' קמז) explained that mikrai kodesh means making it holy and respecting(kovod) it with clean clothing, and making it pleasurable with the pleasure of eating and drinking.
There is another option (the rambam) that mikrai kodesh means to make it holy by not doing prohibited work,
but the honor for shabos and yom tov and the pleasure on shabbos is rabbinic [not from the Torah but only taught by verses in the Prophets] (but (ב) the pleasure and joy of yomtov with food and drink is biblical...) Anyway (even if the second opinion above is correct, it is necessary to be very careful in these things, [because] 1. rabbinic decrees are stricter than biblical ones, and 2. Anyone that makes Shabbos be a time of pleasure his reward is clearly brought in kabolo [what is handed down] (Isiah 58.14) "Then, you shall delight with the Lord..." And in the words of our Sages (shabbos 118.2) he is forgiven for all his sins and he is protected from the judgement of gehenom