Your slogan should be 'Rest assured. My wife will do it.'
The man, Walter Gruenby, is planning to run for political office, but he's very lazy (indicated by his relaxing in the yard), compared to his wife. While the man has aspirations of political grandeur, his wife retorts by saying she does everything for him, including potentially doing all of his political work, and he should thus change his slogan. A play on the idea that wives do everything and husbands do very little.