Tapering is not tightening in the level sense. The balance sheet is still growing, just more slowly. Markets nonetheless treat the announcement as the start of the exit, because it fixes the date at which purchases stop and brings forward the expected first hike.
The 2013 taper tantrum is the reference episode: a hint that purchases might slow sent the 10-year yield up roughly 100 basis points over a few months. Central banks have since telegraphed tapers far in advance to avoid a repeat.
Example: monthly purchases of $120 billion are reduced by $15 billion per meeting. That implies eight meetings to zero, so the market immediately prices the first rate-hike shortly after that endpoint.
Related: quantitative-easing, balance-sheet-runoff, forward-guidance, term-premium