Why would someone suffering from Alzheimer's play checkers well but have a hard time holding a sensible conversation?

Because Alzheimer's effects explicit memory and leaves implicit memory intact

Alzheimer's affects the brain as a whole, but it is not uncommon for someone suffering from it to be able to remember how to do certain things that they could do well before the condition set in. Because they hadn't had the particular conversation before or may be confused about what is being spoken about, they may not be able to have a sensible conversation.