You can picture hasten as a person suddenly walking faster towards somewhere they need to be - the pace picks up, and the place they are heading for is already in view. Everything else the verb does keeps that quickened pace and changes what is being quickened. First it is your feet. Then it is your mouth, when you get a sentence out at once instead of waiting. Then it is something outside you altogether: an ending, a decision, a recovery, pulled forward in time.
Hasten was formed from the noun haste plus the verb ending -en, and it started life as an intransitive verb - that is, as somebody moving in haste, with no object after it. The 'make something else happen faster' use, which is the one most readers meet today, came from that older picture of a person hurrying.
hasten 是 haste（匆忙）加上動詞字尾 -en 造出來的，最早是不及物動詞，後面不接受詞：講的就是人自己在趕。今天最常見的「讓別的事情變快」是從那個「人在趕路」的畫面延伸出來的。
Wiktionary entry for hasten, supplied in the evidence packet (source_ref kaikki:3f6f69a190036ae4:440247181:20998): 'Originally intransitive, from haste + -en (verbal suffix).'
From the picture
讓別的事變快或提早
這裡加快的不再是自己，而是身外的一件事。句子多了一個受詞，主詞也常常不是人，而是一個情況或原因：寒冬讓他們提早離開，大雨讓運動會提早結束。可以想成把原本推著自己走的那股勁，轉去推一件事的進度。
verb · B2
To make something happen faster than it would have happened on its own, or to make it happen at an earlier time than it otherwise would.
加速；使提早
說話時趕緊補的一句
這一格是從上面的 hasten to do something 固定下來的一句話。要趕緊做的那件事，就是把後面那句補充說出口。它已經不能拆開來理解，只活在 I hasten to add 這個形狀裡，主詞固定用 I。
verb · C1
Said when you quickly add something to what you have just said, because you think it could be understood in the wrong way.
我得趕快補一句
verb · C1
To go somewhere quickly. This use nearly always has words telling where you are going: home, to the door, along the road.
急忙前往
verb · C1
To do or say something quickly and without waiting, especially to explain, add or make something clear straight away.
趕緊（說／做）