Sieh beim Überqueren der Straße nach links und rechts.

English Translation

Look to left and right when crossing the street.

Why is this genitive?

The english “when crossing the street” really is short for “when you are crossing the street”.
But “beim Überqueren der Straße” is literally more like “during the crossing of the street”, so “of the” is a hint to use the genitive in german.
This is a common pattern: <nominalized verb> + <noun in genitive>
Example:
Das Schweigen der Lämmer - The silence of the lambs. (or more literally “The keeping quiet of the lambs”)

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“schweigen” is an interesting choice for an example here, because we actually don’t have a verb for that in English. But you still explained that quite well.

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