Saitama's power is reversing cause and effect on attack and defense.Like Lancer's Gae Bolg from Fate/stay night, he does something; and reality warps so that the effect is that his punch crushes people, or their attacks on him aren't as effective; regardless of his physical strength. Similar to Lancer; the ability has an Equivalent Exchange on his luck. Unlike Lancer, he doesn't get hurt; however people never recognize what he does. (Because they can't rationalize the shift in cause and effect.)
Saitama is a product of the same phenomenon as the random mysterious beings that are mutated human beings.The series has introduced examples of human beings that have repetitively performed certain tasks that cause them to transform in a fitting fashion. A man who ate too many crabs became Crabrante, a giant crab man. A person shadowboxed a ceiling light pull cord so much that he became Personification of a Light Pull Cord, a powerful and fast boxing giant. These beings were still people, as they spoke normally and the first example even engaged in a somewhat civil conversation with Saitama at first passing. They were influenced by their power, however. One sought revenge for being drawn upon, and the second desired to test his strength against objects and people. While these beings were people influenced by their transformations to the extent that they became disruptive to the lives of others, Saitama is different from them only in that he is heroic and doesn't desire to harm innocent people with his power. His extremely powerful punch and generally superhuman fitness is a showing of his desire for strength that he sought through training. He trained so much and so hard that whatever phenomenon caused the changes in the other two beings mentioned happened to him as well. As the others transformed into monsters, Saitama transformed as well; he simply became bald. Thus, he is literally a monster fighting other monsters, but he is a monster that retains most of his original human form.