Unaccountably means that something happened and nobody can explain why — and the speaker finds that strange. You put it next to the surprising thing: 'she unaccountably burst into tears', 'the train was unaccountably late', 'he felt unaccountably happy'. The word does two things together: it reports the fact, and it admits that no reason can be given for it. That admission is the whole point. If you know the reason, you do not use this word. It is a formal, rather literary word, common in novels and news writing and rare in speech; in conversation people just say 'for some reason' or 'no idea why'. A very common place to meet it is before a feeling that arrives without cause — unaccountably nervous, unaccountably cheerful, unaccountably drawn to someone — because feelings are exactly the things that show up without permission. Note the older, stricter sense of the root: 'to account for' something means to explain it or to answer for it, and to be unaccountable can also mean 'not answerable to anyone', as in an unaccountable government. That second meaning belongs mostly to the adjective; the adverb unaccountably almost always means 'mysteriously', not 'without responsibility'.

莫名其妙地、說不出原因地。文章裡常見，口語通常直接說「不知道為什麼」。注意它帶著「講話的人也覺得怪」的意味，不是單純的「奇怪」。