In The Bible, Why Did Joseph Arrest His Brothers After He Framed Them For Thievery?

There were lots of reasons.  He wanted to teach his brothers how it felt to be placed in the situation they had placed him.  He desperately wanted to see his full blood brother, Benjamin.  He also wanted evidence that their attitudes had changed since they sold him off as a slave from jealousy.

Their conversation at table made him realise that they did feel remorse for what they had done and set the stage for him to offer them forgiveness once he made himself known to them.

Joseph, a Hebrew/Jewish young man, had become a slave in Egypt after his 11 older brothers by a different brothers had sold him to a camel caravan then dipped his coat of many colours into sheep's blood then gave it to Jacob/Israel, their father to prove that Joseph was dead.  (His father's name had been changed by God from Jacob to Israel later in life before the tribe of Israel went to Egypt during a famine and all of them, the Israelites/Jews, became slaves of one of the Pharaohs of Egypt, not the one who invited the tribe of Israel to live in Egypt.)

Years later during a famine, Jacob/Israel sent 11 of his sons to Egypt to, hopefully, buy grain and other food from the Pharaoh who had saved up wheat and other food crops during the previous years of great abundance.  His youngest son, Benjamin who shared the same mother as his son whom he believed was dead, stayed home with his father who loved him beyond the love of any father to his son.

When the 11 brothers arrived in Egypt, Joseph was in charge of the distribution of the saved crops and food.  He recognized his brothers but they did not know him.  Joseph did not allow them to know who he was, however, he went off by himself and wept out of joy that they were still alive, their father was alive and there was a young brother of his who was born to his mother.

Later, he arrested them as spies, held the oldest brother, Simeon, in jail as hostage until the others returned bringing their youngest brother with them.  They did as they were demanded.  When Joseph saw Benjamin, he went off by himself and wept again.

Later, when they left with Benjamin, he had his officers place his own silver cup in Benjamin's sack of wheat so he detained his young brother until the others returned with their elderly father, Israel formerly called Jacob.  They begged him to not put Benjamin in jail because it would break their father's heart.

He had framed them because he wanted to see his father and invite his family to live in Egypt during the famine so he could take care of them.

(Genesis Chapter 42 and 43)