Sentences don't ordinarily occur in a vacuum: context is usually critical to understanding.
So with auxiliary inversion in the following clause is usually employed with the sense "thus"; this is an anaphor, referring to an action mentioned immediately before the clause. It usually carries the additional sense "too, also". Here's your sentence in context:
But a few days later they could hear a bouncing noise outside the shed; bump-bump it went against the wall, and the Wheelbarrow knew what that meant. So did the Old Ball; it lay there sighing through its crack all day long.
So did the Old Ball means that The Old Ball, too, knew what that meant.
Sighed through its crack is also explained in context. Just three short paragraphs earlier we read:
“Look, there’s my ball!” she shouted, and hugged the Ball tightly. But it sagged and made a hissing noise because it had a split in its rubber tummy.
The crack is the split in the ball; as JavaLatte says, the author treats this as an "anthropomorphic mouth" through which the ball sighs.