Realism is defined by a focus on "reality" as it can be expressed by our own perceptions, and by a focus on present-day, contemporary settings and events, instead of on the extraordinary (Wikipedia).
Saul Bellow's works are all set in the present-day during which he wrote them, and they all deal with regular people dealing with their own lives. His most adventurous novel, Henderson the Rain King, is still rooted in a rich heir's dissatisfaction with his own purpose, and his well-known existential novel, Herzog, still leads to a real-world conclusion stemming directly from the main character's actions.
While he specifically rejected the label of Realism, Bellow shows clear influence from the genre; his novels rarely addressed politics, instead focusing on people and their problems. In showing the intellectual side of common life, Bellow addressed larger themes while still digging into the roots of motive and action. Because of this focus, Bellow was able to straddle the line between Realism and Transcendentalism, which also had a great influence on his work; however, he tended more towards the realistic portrayal of people and emotion in interacting with each other, instead of their reaction to the world around them.