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Blast attack will make weapons that have blast status on them better. You need a blast weapon for it to work.
The way blast works is that as you attack with your weapon you build up like a "blast meter". Depending on the monster you need more blast build up. Once you reached the limit, it will trigger an explosion that does a set amount of damage. Blast attack on the armor makes the build up quicker and the explosion do more damage.
Blast works similar to status ailments, while you attack the monster with blast weapon it will build up and once it reach the max "capacity" it will explode where you've hit the monster dealing a lot of damage.
Blast, like all statuses, works best on faster attacking weapons like the sword and shield, dual blades and insect glaive. If you're using a weapon like the great sword you won't get much benefit from blast boosting skills.