This may be a silly question but it’s been difficult to Google, mainly because it’s such a common word. The etymology is straightforward:1867, from breath + present participle of take (v.). Phrase to take (one’s) breath away with astonishment or delight is from 1864. Breathtaking (n.) “act of taking breaths or a breath” is from 1620s. Related: Breathtakingly.
But what is happening on a physical level? We all experience it. The sudden little catch in one’s breath when seeing something (or someone) who is well, breathtaking, for want of another word. Is it just a sudden quickening of the pulse causing a demand for more air?