Learn Dutch: Ik schaak niet meer. - I do not play chess anymore.

In English, you say "play chess", using a verb and a noun. In Dutch, schaken is a complete verb on its own meaning "to play chess". Here, schaak is the conjugated form for Ik, so Ik schaak simply means "I play chess".

Why do we use niet meer here instead of geen ... meer?

You use geen to negate indefinite nouns (like "a book" or "water"). Even though "chess" is a noun in English, remember that schaak is a verb in this Dutch sentence. To say you no longer do an action (negating a verb), you must always use niet meer.

Could I say Ik speel schaak niet meer instead?

Dutch speakers almost always use the verb schaken rather than combining a verb and a noun like schaak spelen. While een potje schaken (to play a game of chess) is a common phrase, for the general hobby, just stick to the single verb schaken.