"in the end, finally; after all: In the end they shook hands and made up."
I can't find any idiomatic meaning for "at the end", but this phrase makes perfect sense if taken literally: the end of something.
There are some similar but not exact phrases: "at wit's end" -- near insanity or desperation; "at land's end" -- on a shore with no land visible across the water; etc.
"Into the end" is not used, but there's an archaic phrase "unto the end", modern "to the end", which means "until the end", and is used in a metaphoric sense: until the very end, until everything happens that fate decrees shall happen.
These prepositional phrases cannot really be explained. That's why they are called idioms.
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