It's just self delusion, and here's how it can be demonstrated:
The only prayers that are ever "answered" are prayers for events that are possible without a god: Getting a new job, recovering from a recoverable illness, finding a life partner, finding your keys, etc. All these are events that are very much within the realm of possibility and so they will inevitably coincide with prayers asking for them, giving one the illusion of answered prayers.
The only prayers that are never answered are prayers for events that are naturally impossible but should be possible with a god: An amputee regrowing a limb, a week old corpse coming back to life, a mountain being literally lifted up and transferred to the sea. All these are events that are impossible with man but should be possible with god. But prayers for such things are never answered. Why? Because there is no god listening to prayers.
Possible events happen and they will inevitably coincide with prayers for them. Impossible events won't happen and will never coincide with prayers for them. There is no god listening to and answering prayers. It's all self-delusion through coincidences and confirmation bias.