Before us is Mishnah 4 in chapter 4 of Maseches Shevi'is, which deals with the law of one who thins out olive trees in the Shemittah year. To "thin out" means to remove some of the olive trees that are planted too close to one another, since trees need adequate space between them in order to grow properly. One who thins out olive trees, then, is someone who uproots several of the olive trees in his grove that are planted too densely.
The dispute between Beis Shammai and Beis Hillel:
"Beis Shammai omrim: yagom" - Beis Shammai are concerned that one who does this during Shemittah will appear to be preparing the ground from which he uprooted the trees for planting. He must therefore cut down the trunk in a way that leaves a bit of the trunk and the roots in place, thereby making it clear that he has no intention of using the ground beneath for planting.
"Beis Hillel omrim: mesharesh" - In the view of Beis Hillel there is no need for such a concern, and he may uproot the tree down to its roots, even though to the eye it appears as though he is preparing the spot for planting.
Nevertheless, "u'modim b'machlik ad she'yagom" - Beis Hillel agree with Beis Shammai in the case of one who "levels," that is, when he uproots several trees in one spot and creates there an open, cleared area. In such a case he may not uproot the olive trees unless he leaves part of the roots in place, so that it will be clear that he does not intend to plant there.
Defining the terms:
"Eizehu ha'meidal? Echad o shnayim" - One who uproots one or two trees.
"Ha'machlik? Shloshah zeh b'toch zeh" - One who uproots three trees adjacent to one another, thereby creating a single open area. This is the case in which Beis Hillel agree with Beis Shammai that part of the roots must be left in place.
"Bameh devarim amurim?" - In what case was this said?
"Mitoch shelo" - Everything stated above applies when a person uproots the trees from his own field, where there is a concern that he will appear to be preparing the ground for planting.
"Mitoch shel chaveiro - af ha'machlik yesharesh" - When one uproots from his fellow's field, even one who creates a continuous open area may uproot the roots as well, since in another person's field there is no concern that he will appear to be preparing the field for planting.
In summary: In this Mishnah we learned the law of thinning out olive trees during Shemittah: according to Beis Shammai one must cut down the tree and leave part of the trunk and roots, because of the appearance of preparing the ground for planting, while according to Beis Hillel it is permitted to uproot completely. Beis Hillel agree with Beis Shammai in the case of "leveling" - uprooting three adjacent trees, which creates an open area - whereas "thinning" is the uprooting of one or two trees. And all of this applies in his own field; in his fellow's field, even one who levels may uproot completely, since there is no concern there about appearances.