Refers to the act of pretending not to know something, often in a playful or evasive manner. Commonly used in the phrase とぼけを決め込む (to brazenly play dumb).
A more colloquial and slightly stronger verb meaning 'to feign ignorance' or 'to act innocent', often with a nuance of shamelessness.
Etymology
Derived from the verb とぼける (to play dumb). The kanji 惚 is associated with being absent-minded or infatuated, but the exact historical derivation is uncertain.