What does "I have eaten myself stupid" mean?

It is a variant of the following idiomatic expression which is used to suggest that you did something (drink, laugh. etc.) to an extreme point:

Drink, laugh, etc. yourself silly: ​

• to drink, laugh, etc. so much that you are unable to think clearly or behave with good judgment:
  • I laughed myself silly at his jokes.

Cambridge Dictionaty

"Eating yourself stupid" means your body is reallocating energy to digestion and taking it from the brain, to the point your cognitive processes are compromised.

If you have eaten a sufficiently large quantity of food then the body must devote some proportion of your "immediate" energy to digestion. This means taking it away from the brain, so you feel lethargic and experience a period of low cognitive power/ability.

A similar thing happens while doing long periods of exercise, like a 4+ hour bike ride. The gut takes power away from your legs and you go a bit slower, which is why easily-digested gels and simple foods work better. Being tired takes power away from the brain too, so riders in extreme power deprivation states (i.e. 24+ hour rides) can start acting very stupid, like "getting off their bike and lying in the roadway for a nap" or turning around and searching for their mum.

It sounds like the British Formula One driver, Jenson Button, is careful about his weight. He has to watch the calories because not only is he a driver but also a sportsman.

When people have to follow a strict diet regime for any extended period, it soon becomes frustrating and boring. The moment Button terminates his F1 contract, he admits he will overindulge by eating those very foods which were restricted or expressly ‘forbidden’. In this frenzy of eating, he'll most likely overeat, and feel full. Only a stupid person would want to reach that uncomfortable stage; either that —or— as suggested by @Tom Fenech, risk becoming stupid in the process, i.e. eat oneself stupid.

Similarly, people say they went on a binge or they binged on junk food until they felt sick.