וַֽיְשַׁלְּחֵ֛הוּ יְהוָ֥ה אֱלֹהִ֖ים מִגַּן־עֵ֑דֶן לַֽעֲבֹד֙ אֶת־הָ֣אֲדָמָ֔ה אֲשֶׁ֥ר לֻקַּ֖ח מִשָּֽׁם׃
האָט גאָט דער האַר אים אַרױסגעשיקט פֿון דעם גאָרטן פֿון עֵדֶן, צו באַאַרבעטן די ערד, װאָס ער איז פֿון דאָרטן גענומען געװאָרן.
Chizkuni
וישלחהו מגן עדן, “He expelled them from Gan Eden”. He was punished in a manner similar to that of a person killing another person through neglect rather than with intent to kill, i.e. he is exiled to a city of refuge. [the whole earth other than Gan Eden in this instance. Ed.] The reason why this penalty was appropriate was that at the moment when Chavah and Adam had eaten from the tree of knowledge they had not yet possessed the knowledge of good and evil which would have made them deliberate sinners.
Kli Yakar
To work the ground from which he was taken. It is not for nothing that the text mentions from which he was taken. I also found (in Midrash Shochar Tov, Psalms 92) that it states that Adam settled on Mount Moriah. From where did our sages learn to say this? Additionally, our Sages (Chullin 60a) state that Adam offered a bull with horns and hooves [as a sacrifice] — what biblical support did they find for this? It appears that the basis for all these matters comes from the phrase from which he was taken — namely, from the place about which it is said an altar of earth you shall make for Me (Exodus 20:21). As our Sages said in Bereishit Rabbah (14:8), “The Holy One, Blessed be He, created him from the place of his atonement, and may it serve as his atonement.” This refers to Mount Moriah, where God sent him to work the ground and build from it an earthen altar to offer sacrifices for his atonement. Since he was taken from that earth, and it was the gateway through which he passed — for the earth provided him with coarse and gross material which caused him to sin, as explained above in the verse tree bearing fruit (Genesis 1:11) — therefore, in the place that caused the sin, there shall be his atonement. For that place, meaning that earth, caused him to sin; therefore that earth is obligated to assist in his atonement through his working it to make from it an altar upon which to offer a bull with horns and hooves. This is a proper reason for the commandment an altar of earth you shall make for Me, for through the very thing that caused the sin, he will repair what he corrupted. And even according to the opinion that states that His dust was gathered from all the earth, this is its explanation: Since the place where it says an altar of earth you shall make for Me is where the Foundation Stone is located, and from there the entire world was founded, the dust taken from the center of the world is as if it were gathered from the entire world. This also resolves what is said The land upon which you lie, I will give to you (Genesis 28:13). Rashi explains that God folded all the land beneath him. It was not necessarily folded, but rather, since he lay on the navel of the earth and its center, it is as if he lay upon all the land.
Sforno
וישלחהו, He expelled him. G’d ordered Adam to leave the garden. The expression is used in the same vein as Exodus 12,33 למהר לשלחם מן הארץ, “hastening to send them out of the country.” (Pharaoh)