This is saying that a president, or any person for that matter, can only handle so much at a time.
The outdated custom of eating peas with a butter knife took having an awareness (however slight) of the position, status, and possible movement of each pea in order to successfully eat them without dropping one, or all. In the same way, every person has a limit to what they are able to effectively manage, carry emotionally, or otherwise deal with, before their peas, so to speak, roll off of their knife.