"had packed it up to the walls"
The image is that there were so many pupils in the school, they were packed in like sardines in a can -- they were even pressed up against the walls. This is, of course, an exaggeration -- an example of what we call "hyperbole". In fact, the meaning is simply that the school was running at full capacity.
"stuffed a baked potato in my pocket"
I don't know if this was a tradition, but I guess this was intended to help keep Lee warm on the way to school. It could also have been his lunch.
"went skating and skidding around me"
This simply means they skated and skidded around Lee. The use of the construction "to go" + -ing form helps to reinforce the idea of a lot of children all surrounding Lee, skating and skidding around in a confusing manner.
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