This mishnah addresses the question of at what point a betrothed young woman leaves her father's authority and enters her husband's authority, and from when those items listed in the previous mishnah pass over to him.
The text of the mishnah:
"L'olam hi bireshus av ad shetikanes lireshus baal l'nisuin" - the young woman, and likewise a minor girl, remains in her father's authority until she enters her husband's authority for the sake of marriage. Once she has entered the chuppah for the purpose of marriage, the authority passes from the father to the husband, and all those items listed in the previous mishnah now belong to the husband.
The father handed her over to the husband's agents:
"Masar ha'av lishluchei habaal - harei hi bireshus baal" - even though she was not handed over to the husband himself but only to his agents, she has already passed into his authority, and those items listed in the previous mishnah now belong to the husband and not to the father.
In the Gemara the Amoraim disagree as to whether this handing over is effective even regarding eating Terumah, since a woman who marries a kohen may eat Terumah only after the marriage:
Rav: she is like his wife for all the matters mentioned in the previous mishnah, except for Terumah.
Rav Asi: the handing over to the agents is effective even regarding Terumah.
Traveling along with the agents as opposed to an actual handing over:
"Halach av im shluchei habaal" - the father accompanies the husband's agents: "harei hi bireshus av" - she is still in her father's authority.
"O shehalchu shluchei ha'av im shluchei habaal" - even when the father's agents accompany the husband's agents, she still remains in the father's authority.
"Masru shluchei ha'av lishluchei habaal" - once the father's agents have handed her over into the hands of the husband's agents, "harei hi bireshus baal" - she is in the husband's authority from that moment on.
In summary: we learned that a young woman remains in her father's authority until she enters her husband's authority for marriage, and that even handing her over to the husband's agents transfers her to his authority (and Rav and Rav Asi disagree as to whether this applies even regarding eating Terumah). We also noted the distinction between accompanying and handing over: as long as the father or his agents accompany the husband's agents, she is in the father's authority, whereas once she has actually been handed over to the husband's agents, she is in the husband's authority.