One way to hold flatly: picture something pressed down until the whole of it touches — a sticker smoothed onto a book, sand running level to the sea. Nothing lifted, nothing sticking out, nothing to catch hold of. That is the plain physical use. Move the same picture onto speech and it works twice over. A no said flatly has no gap in it, nothing to argue with, nothing to pull at. A sentence said flatly has no rise and fall in it, so there is nothing in the voice for a listener to read. One surface; two things that can be made level — an answer, and a voice.
From the word’s history
Wiktionary gives flatly as flat plus -ly, tracing flat back through Middle English to Old Norse flatr and Proto-Germanic *flataz. So the level surface is the word's starting point, and the speech uses are that surface carried over.
Wiktionary 把 flatly 拆成 flat 加 -ly，並把 flat 一路追到古諾斯語 flatr 和原始日耳曼語 *flataz。所以「平面」是這個字的起點，用在說話上的兩個意思，都是把這個平面搬過去用的。
Wiktionary, 'flatly' (adv) etymology tree, supplied in evidence packet (source_ref kaikki:3f6f69a190036ae4:1351097204)
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