Take 人 and open the arms: 大 is a person taking up maximum space — the body's own gesture for 'this big!'. The script's size vocabulary is anthropomorphic: bigness is a human stretching.
大 fronts China's proudest nouns: 大学 university ('great learning' — also the title of a Confucian classic), 大家 everyone ('big family'), 大人 adult. Ordering 大份 big portion is optimism; being called 大方 'big-square' means generous and poised.
Arms flung wide — 'it was THIS BIG!'
大夫 read dàfū was an ancient official rank, read dàifu it's a doctor — one character, two pronunciations, two careers.