Sword Art Online: Integral Factor Wiki

Burst skills are a unique type of 4 star skill easily identifiable by the purple rarity stars, as opposed to the normal gold ones for most skills. The burst skills themselves function as normal 4/5* skills with different strengths and effects on each. As well as having a stronger burst attack usable by chaining certain skills before using the burst skill. When used as a burst attack they get a sizeable strength boost as well as a stronger bonus effect and temporary invincibility while the burst attack is active. When used as a normal attack, they have average cool downs and function as you would expect from most skills. When used as a burst attack, the cool down is a bit longer.

To use a burst skill as a burst attack you must first level the skill to 70. You don't need to transform it, recommended maybe, though not needed to use as burst, just get it level 70.

To activate the burst attack you need to use a chain of 3 skills in a row of specific types/elements, then the burst skill, not as a switch, just the normal skill. Each burst skill has its own chain you need to perform, and you have around 8-10 seconds between each skill to keep the chain going. Upon using the first skill in the chain a bar at the bottom of the screen will show the time remaining for the current step, as well as an icon with the currently needed type/element to continue the chain, with multiple icons if more then one burst skill is equipped. If the bar empties the chain reverts to the previous step. It will also break the chain of you use the wrong type of skill for the chain. (For instance, using a wind skill Instead of a fire skill)

As an example for the Black Pioneer Kirito skill you need to use a slash skill, then a fire skill, then another fire skill, then use the burst skill.

Here's a quick example video of a burst skill in action by Redditor YangLionSpirit.

It can be any slash skill first, including one from a separate weapon. Same for the fire skills. You can use rapier moves to charge a sword burst or a spear to charge dagger burst etc, switching between the two weapon sets you can equip. (Wether swapping weapons mid battle is a good tactic or not is another issue) For kirito it just has to be a slash skill, then a fire skill, then another fire skill (or whatever it is for the burst your using), then the burst skill. Also your switch can be used as part of the chain to save on some sp or get an extra use out of a skill, just not to activate the final burst attack.

If you have the right skills to complete the chains, you can equip multiple burst skills and use each of them. However making all the chains work right can get quite complex and possibly too much effort for the return. Using one on each set is fine, but two burst skills on one weapon or trying for 3 or more burst skills would get to be not worth the trouble.