‘ And Boaz said to the elders, and to all the people, “You are witnesses this day, that I have bought all that was Elimelech's, and all that was Chilion's and Mahlon's, from the hand of Naomi. Moreover Ruth the Moabitess, the wife of Mahlon, have I purchased to be my wife, to raise up the name of the dead on his inheritance, that the name of the dead be not cut off from among his brethren, and from the gate of his place. You are witnesses this day.”
By accepting the shoe Boaz would be declaring to those present that he was accepting the responsibility for redeeming the land and begetting a son through the wife of the deceased in order that Mahlon's name might be perpetuated (the son could be called ‘ben Mahlon'). And he now therefore turned to the elders whom he had summoned as witnesses, and to all the people gathered in the gateway, and called on them to be witnesses to the fact that he had bought the land that had belonged to Elimelech, which had been inherited by his two sons, from Naomi, Elimelech's wife, (the eldest surviving near relative) and that with it he had bought the privilege of begetting sons through the wife of Mahlon in order to preserve the name of the family of Elimelech and Mahlon. That land would eventually devolve on the son produced by Boaz and Ruth, so that the name of the family of Mahlon would have been preserved in Israel. He would be known as ‘Obed ben Mahlon ben Elimelech' as well as ‘Obed ben Boaz'. Note the repetition of ‘you are witnesses this day'.
Being ‘cut off from the gate of his place' probably indicates being removed from the permanent genealogical records of the place which was his home city. Where a family died out there would be no purpose in maintaining the records. The city records would probably be held in the gatehouses.