The 大 was used like 大いに and 切 was being used to mean something like "to urge or compel." It didn't mean "cut."
That might sound strange, but 切 was taken to mean "to urge or compel" in other words as well, like 切迫. It's important to remember that kanji are really old and often have lots of tangentially related meanings beyond what we think of as one English word we associate with each them.
It's also worth noting that although the kanji are borrowed from Chinese, the word 大切 is a compound created in Japan.
From that starting point of something like "needing great urgency", it got generalized to something more like "extremely important" or "critical." Further changes to the nuances aren't too hard to imagine after that. This happened over the course of more than 1000 years.
Origins of this word?
Some kanji pairings are really obvious (like map - 地 ground - 図 - map/drawing/plan) but does anyone know how 大切 came to mean 'important'?