Why is the definite article used here?
Solution 1:
As mentioned in the comments "run under the bed" is idiomatic. This is actually quite common in English, and the object used with the definite articles in these idioms is usually a particular one, but not necessary particularly identified. Some more examples:
"I saw a car parked on the shoulder." Which shoulder?
"I heard someone walking up the stairs." Which stairs?
"She took my phone and threw it out the window." Which window?
"He was just standing in the corner, not mingling much." Which corner?
"I looked her in the eye, and lied through my teeth." Which eye?