I don't think it's a metaphor.
It informs you of the background of the story.
It's 장마철, so it's summer's heavy rainy season, which is usually early August in Korea.
보충수업 means it's the School setting. The protagonist might be a teacher? or a student? Probably the teacher at the school's nursing office as it says from the title.
보충수업 기간 used to be typical for Korean students as they were required to come to school during the summer vacation to study. (It's like an extra summer programs in America)
신발장 냄새가 진동했다 --> To me, it gives an impression of very damp air. 신발장 at school are usually made of wood, which smells a lot when wet.
As a whole, it reminds me of my school days in summer back in the past.
So, it's more of an imagery + introduction with background information. Not metaphor to me.