Why was he able to escape from the palace and why was it suddenly necessary to escape?

The palace had rusty old locks that were easy to break. Equality really wanted to get his glass box to show to the World Council of Scholars and prove what he has done.

The old locks and lack of guards in the Palace of Corrective Detention made it easy to escape. it was necessary for Equality to escape so he could show his experiment to the World Council of Scholars.