not cut it
- to not be good enough to do something:
e.g. I don't get it why such a talented guy just can't cut it.
e.g. If she can't cut it, we might need to get someone else to take over the case.
e.g. Players who can't cut it soon quit the team.

Can anyone briefly explain why this phrase has such a meaning?

I think the expression is older than I am.

The only related expression I know is "Can't cut the mustard," and the variation, "To old to cut the mustard." I'd have to do a search to find what it has to do with mustard.

In professional big bands, the rule was "Cut, or no notice." If you couldn't play the part, you were fired with no notice, i.e., immediately.

I think I've seen threads on this subject before. You might try our local search engine.
If you want to know how it came to mean what it means, I can offer this explanation:

if one can't cut it, or can't cut the mustard, they're like an old knife with a blunt business end, and hence they can't do any work and they're useless.
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Makes sense.
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