“he barely ate dinner”
Barely has a negative sense meaning almost not. He barely ate dinner means he ate very little. Tina is worried about Frank's loss of appetite.
A2-56 / CEFR A2 / EPISODE 56
Explaining the course of symptoms
SceneAt a clinic. Frank has stomach pain; a worried Tina accompanies him.

WORDS & CONTEXT
Barely has a negative sense meaning almost not. He barely ate dinner means he ate very little. Tina is worried about Frank's loss of appetite.
Besides an insect, bug can informally mean a minor infectious illness. A stomach bug is a gastrointestinal illness. Compare catch a bug, meaning to pick up an illness.
Check on someone means to see whether they are all right. Unlike simply examining something with check, it focuses on a person's or situation's well-being.
"What a relief." Frank is relieved not only that it does not seem to be serious, but also that Tina will not be calling him in the middle of the night, so he can sleep.
Write ... down means to put something in writing, and sign means to add your signature. That refers to the doctor's instruction just before: don't call him during the night; let him sleep. The punchline is that Frank even asks for the doctor's signature so that worried Tina will follow it.