The core meaning of “espouse” is “marry.” When you espouse an idea or cause in public you are proclaiming that you are wed to it; you are promoting it as yours.
When you expound an idea you are explaining it. Theoretically you could expound an idea that you don’t personally espouse. “Expound” was traditionally used mainly to refer to detailed examinations of complex or obscure systems of thought, but it is most often used today to mean “to speak at length about” and frequently occurs in the phrase “expound on”: “the senator expounded on his love for the traditional family farm.”
Sometimes in such contexts it would be more appropriate to use “expand on,” which means “to speak at further length about.” “Expand” in this sense lacks the systematic analytical connotations of “expound.”
You never “espouse on” an idea; you just espouse it.
The Week's End Extra from the Archives: "Whits and pieces . . ." (April 21, 2011)