Learn Swedish: Visdomen delades av den gamla mannen. - The wisdom was shared by the old man.

The word visdomen is built from the adjective vis (wise) and the suffix -dom, which creates abstract nouns related to a state, condition, or domain (very much like "-dom" in the English "wisdom" or "freedom"). Nouns ending in -dom are almost always en-words, which is why it takes the definite ending -en to become visdomen (the wisdom).

What does the -s at the end of delades do here?

It creates the passive voice. The active past tense is delade (shared), but adding the -s makes it delades (was shared). In Swedish, the s-passive is very common, and it is usually paired with the preposition av (by) to indicate who actually performed the action.

Why does the sentence use den gamla mannen instead of just gamla mannen?

This is the Swedish "double definiteness" rule. When you have a definite noun like mannen (the man) and you add an adjective (gamla) in front of it, you must also add the freestanding definite article (den for en-words). So literally, the structure is "the old the-man".

Could I write den gamle mannen instead of den gamla mannen?