We construct an intersection product on tropical cycles contained in the Bergman fan of a matroid. To do this we first establish a connection between the operations of deletion and restriction in matroid theory and tropical modifications as defined by Mikhalkin in [Proceedings of the International Congress of Mathematicians, Vol. II, European Mathematical Society, Zürich, 2006, pp. 827--852]. This product generalizes the product of Allermann and Rau [Math. Z., 264 (2010), pp. 633--670] and Allermann [Tropical intersection products on smooth varieties, J. Eur. Math. Soc. (JEMS), 14 (2012), pp. 107--126] and also provides an alternative procedure for intersecting cycles which is not based on intersecting with Cartier divisors. Also, we simplify the definition in the case of one-dimensional fan cycles in two-dimensional matroidal fans and give an application of the intersection product to realiability questions in tropical geometry.