The whole point of this joke is that it is a parody of the oft-heard English saying, "to wear the pants in a relationship"; the cartoon is, amongst many things, mocking the idea that there is, can be or should be a "dominant" or "most important" part of any relationship, or indeed in any human collaboration.
"To wear the pants in a relationship", in my experience, is used in English most often in the following three contexts: (1) to be (literal meaning) the male in a heterosexual relationship or, more generally, (2) to be (metaphorical meaning) a "dominant" partner or even (satirical meaning) (3) to have an overswollen sense of one's individual importance in a collaboration and probably a disregard for other people's contributions to that collaboration.
Usages (2) and (3) are often alike when used in the third person, in that they both often have a strong whiff that the "pants wearer" is disrespectful of and lacking acknowledgement for the others who contribute to the relationship, be it a couple or otherwise.
So, although one can't be exactly sure what a cartoonist is getting at, a likely implication of what's going on here is that the man is trying to "pull rank" (i.e. claim authority to have the final say on a decision), possibly because he happens to be the person who has the paid job in the family. The lady is making the point that it takes waaay more than money earning to make any endeavor to work, especially an emotional relationship. She is "wearing the brains", in other words, she is constantly thinking and planning how how family finances are going to meet the couple's needs and avoiding financial disaster, which is no trivial job. Indeed, her expression and the poignant paraphrasing of the dictum probably says she feels severely overstressed and overladen at having to be the one who always has to take this responsibility.