The Clar covering polynomial of molecular graphs is a method to characterize the electronic structure of conjugated systems. By studying the Clar covering polynomials of plane bipartite graphs, the resonance theory of related molecular graphs and their properties can be thoroughly investigated. Based on the theorem related to Clar covering polynomials of plane bipartite graphs, the method of generating functions is utilized to compute Clar covering polynomials of plane bipartite graphs. Recurrence relationship for Clar covering polynomials of a special class of graphs are derived. In turn, explicit expressions the Clar covering polynomials of two classes of catacondensed plane bipartite graphs are computed using the generating function method. On the Clar covering polynomials of plane bipartite graphs, it is possible to understand the electronic structure of chemical molecules, predict their chemical properties and reaction behavior, and design new molecular structures.