To "blow" something is slang that simply means "to fail". The exact nuance will depend on context:
Ralph was upset because he was sure he blew the interview, but then later that day they called to say he got the job.
I tried to set the world record for 1000-meter skipping, but I blew it when I accidentally hopped instead of skipped in the last 100 meters.
My kids tried to make my wife breakfast for Mother's Day but they kind of blew it when they fried the toast and toasted the eggs. But she ate it anyway.
There are many ways to use this in a sentence, and for the most part the exact phrasing doesn't matter. "He blew it on packing" is just an alternate way to say, "He blew the packing."
However, it's not the best phrasing because it sounds like the unrelated idiomatic expression to blow (one's money) on [something] which means "to waste all one's money on".
He made a fortune but blew it all on expensive cars and ex-wives.
It's unnecessarily confusing, because in context it makes no sense to spend money on "packing".
Additional note: This should not be confused with the idiomatic expression to blow [someone] away (or for a loop, or out of their seats, or various similar expressions) which means "to impressively succeed".
The audience weren't sure of her when she got up on stage, but once she opened her mouth and started to sing, she blew them away with her talent.
Also not to be confused with to blow away meaning "to completely destroy":
Han Solo: Our position is correct, except, no Alderaan.
Luke Skywalker: What do you mean? Where is it?
Han Solo: That's what I'm trying to tell you, kid. It ain't there. It's been totally blown away.
Exhaustive list if idioms that use "blow"