A person seen from the side with the mouth thrown open — a yawn, or a gasp. From an open mouth came the sense of something missing, and from something missing came debt. As a component on the right it still marks characters about breath and mouths: 歌 (to sing), 吹 (to blow), 欢 (joy).
欠钱 is to owe money and 欠账 is an unpaid account, but 欠 also does quieter work: 欠缺 is a shortcoming, and 打哈欠 keeps the original yawn alive. To say 我欠你一个人情 — I owe you a favour — is to name a debt no money settles.
A person kneeling with mouth wide open — a YAWN, and what is missing is what you OWE.
欠 is a yawn. The same open mouth is why 歌 (to sing), 吹 (to blow) and 欢 (joy, a shout) all end with it.