Bekhorot 58A

כדי שתהא בהמה מצויה לעולי רגלים.

The reason is so that animals will be readily available to those ascending to Jerusalem for the pilgrimage Festivals. The animals are tithed shortly before each of the Festivals so that their owners will refrain from selling or killing them until those times, when they gather them together to set aside the animal tithe. As a result, there are plenty of non-sacred animals ready to be sold, for food or offerings, to those on their way to Jerusalem.

ואף על גב דתנן: עד שלא הגיע הגורן מותר למכור ולשחוט ⁠— ניחא ליה דליתעביד מצוה בממוניה, והדר לזבון וליכול.

And even though we learned in the Mishnah: Until the time designated for gathering arrives, it is permitted to sell and slaughter the animals, which means that there is no obligation to keep the animals until those dates and there is no concern that the owners will sell the animals before tithing them. This is because a person prefers to perform a mitzva with his money, by separating the animal tithe, and only afterward sell or eat those animals.

ואמאי קרי ליה ״גורן״? דטבלה כגורן.

The Talmud asks: And why does the tanna call the time designated for gathering animal tithe goren, which literally means threshing floor? The Talmud explains that it is due to the fact that the arrival of this time renders the halakhic status of new animals like that of untithed produce gathered on a threshing floor, as from the time that produce is gathered on the threshing floor one becomes obligated to tithe it. Likewise, from this stage onward it is prohibited to slaughter the new animals until they have been tithed.

וכמה ״פרס״? פירש רבי יוסי ברבי יהודה: אין פרס פחות מחמשה עשר יום.

The Mishnah states that according to R' Akiva the times for tithing animals are adjacent to Passover, Shavuot, and Sukkot. The Talmud asks: And how much is adjacent [peras]? R' Yosei, son of R' Yehuda, explained that the term adjacent means not less than 15 days.