“Blowing” a vein means that the needle went through it, instead of remaining in the vein where it’s supposed to be. Very easy to do with needles as sharp as they are (those needles are unbelievably sharp - I grazed one across my hand once extremely lightly, and it just sliced me open). It will leave more bruising than a normal stick because the vein has two holes instead of one now, and is more likely to bleed into your tissue a little.
(I blew a vein on a patient once, and mentioned it to them, not realizing that they aren’t up on lab tech speak - the patient was freaked out until I explained to them that their veins weren’t blowing up or anything like that.)