Two are grasping the garment: The mishnah is specifically talking about when the two men are grasping the garment, since they both have a claim to it and neither one has a stronger claim than the other. Because if only one person was holding it, the other person would have to bring a proof with witnesses that it is his; and he wouldn't be believed to take it with [just] an oath.

This one said it is all mine: The Gemara will explain why it taught it twice.

This one swears: The Gemara (Bava Metzia 3a) will explain why [he takes] this oath.

That he doesn't have less than half: The Gemara (Bava Metzia 5b) will explain why this oath was established with this phrasing.

And that one says, "Half of it is mine": He concedes that the half belongs to his friend, so we only judge about half of it (which is what he is claiming). Therefore, "This one says, 'It is all mine,' he swears, etc." like the first verdict (i.e.: the first case in the mishnah). [And regarding] that which they are judging about (the other half), they both take an oath that he has at least half of it, and each one takes his half.

There were two people riding, etc.: It is coming to teach that riding and leading are equal in terms of acquiring an animal that was ownerless.

When they concede: The Gemara (Bava Metzia 8a) will explain [that with this,] it is coming to let us know that [in a case of] one who picks it up for his fellow, it is his fellow who acquires [it].

Gemara - I saw it: Before you picked it up.

He acquired it through sight: From that which it taught, "They divide it."

"He teaches, "All of it is mine": With complete control, since I picked it up first, and you grabbed it from my hands after I acquired it.

But didn’t Rabbenai say: In Bava Kamma (113b) in [the chapter entitled] HaGozel veMakhil.

"Find it," implies it come into one's hands: And even with this, [it states,] "your brother," and not an idolater. And you should not say [that] when [the Torah] excluded idolaters, it was when it did not come to the hands of a Jew; [that] that is when you are not obligated to be burdened by it and return it; but when it comes into his hands, he is obligated to return it, since his lost object is forbidden [for you to keep].

The teacher employed colloquial language: Had the teacher not also said, "It is all mine," I would have said, "What is, 'I found it,' that the teacher taught? It is the language of people that the mishnah employed, and not the language of Scripture - and many people call it a found object from the time of seeing [it].

In general, when he teaches, "I found it": In every place that we taught that one who finds a lost object, acquires it - I would have said that he acquires it from the time of seeing [it]. For no teacher made us understand that one only acquires it by picking it up. That is why we are made to understand this here from the extra [words in the] mishnah.

But doesn’t it teach, "This one," and, "This one": Concerning, "I found it," it taught, "This one said"; and concerning, "It is all mine," it taught, "This one said." And if it was one, this is how it should have taught: This one says, "I found it and it is all mine."

About buying and selling: I bought if from the hands of x. And it is only about buying and selling that we say [that] they divide [it] with an oath. For it is possible to say [that] both of them bought it and that the seller agreed to both of them. But [if] this one said, "I wove it," they would not divide it; for one of them would certainly be a swindler. So it would remain in place until Eliyahu comes.