Vetches of terumah: Beth Shammai says, “They must be soaked and rubbed in purity, but can be given for food in impurity.”
And Beth Hillel says: “They must be soaked in purity, but can be rubbed and given for food in impurity.”
Shammai says: “They must be eaten dry.”
Rabbi Akiva says: “All actions in connection with them [can be carried out] in impurity.”
In order to understand this mishnah we must first explain several rules.
1) Terumah (heave offering) is a portion of produce separated to give to the priests. It is only given from human food and not from animal feed. Vetches, a type of bean, are generally given to animals, but can be eaten by humans in time of need.
2) It is forbidden to cause terumah to become impure. A person who touches terumah must have previously ritually washed his hands.
3) Food can receive impurity only once it has been made wet. While it is dry it cannot receive impurity.
Our mishnah discusses the preparation of vetches of terumah. According to Beth Shammai they must be prepared (both soaked and rubbed) with pure hands, lest the person decide to eat them himself, in which case it would be forbidden to make them impure. However, they may be given to animals by a person with impure hands, since animal food is not really terumah.
According to Beth Hillel, they must be soaked by a person with pure hands, since getting them wet enables them to receive impurity. However, when he rubs them or gives them to an animal it is obvious that they are not intended for humans, and therefore he can do so with impure hands.
Shammai himself is again, stricter than Beth Shammai. He holds that vetches must be eaten dry so that they will not be able to receive impurity. Assumedly, Shammai would agree that a person might also eat them with clean hands.
Rabbi Akiva’s opinion is the most lenient. He holds that vetches are not fit for human consumption and therefore one may do anything with them while his hands are impure. Since vetches are animal food the rules of the purity of terumah do not apply to them.