Theory:

Finally, the last day of school arrived and the elf was free to go. As for homework, there was no more, so he quietly and slyly slipped out the back door.
Explanation:

In the end, it was the final day of school, and the elf  had complete liberty to go as there was no more homework to do. So, without speaking anything, in a secretive way,  he moved out from the back door.
Reference:
National Council of Educational Research and Training (2006). Honeysuckle. Who did Patrick's Homework - Carol Moore (pp. 7-10). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.