Twisted bilayer graphene with a rotational misalignment (twist) angle close to the magic value of 1.1°, features isolated flat electronic bands. These bands form a strongly correlated electronic system that exhibits a range of quantum phases including superconductivity, ferromagnetism and Mott-like correlated insulating states. In this talk, I will discuss how we investigate the properties of magic angle twisted bilayer graphene by probing the local tunneling density of states using scanning tunneling microscopy and spectroscopy. We show that the flat bands get highly deformed when they are aligned with the Fermi level using electrostatic gating. Near charge neutrality, we find a correlated regime featuring a substantially enhanced flat band splitting that we describe within a microscopic model predicting a strong tendency towards nematic ordering. At half-filling of the bands, we observe the development of gaps originating from correlated insulating states. Our results provide a basis for a microscopic understanding of correlated quantum phases in magic-angle twisted bilayer graphene.