I think the idea of davening at kivrei tzaddikim is that they are a makom kadosh, and tefillos recited at a makom kadosh are more powerful.
Also, the zchusim of the tzadikim buried there can provide an extra zchus for your tefilos. We all (chassidish or not) believe that different places have different spiritual realities whether we can feel them or not. The presence of a tzaddik or a tzaddik’s kever gives the place a certain kedusha and therefore the tefilos said in that place are different. The person also becomes a different person as a result of being in a makom kadosh. And therefore, his tefillos are different, and he can connect to Hashem more easily.