To burst into something is to suddenly begin to produce a noise, action, or response. For example, one could burst into song or laughter. Burst into can also refer to an abrupt entry to a place, such as bursting into a room.
Ex: The jubilant choir burst into song as soon as the minister finished his opening prayer for the Easter service.
Ex: The police officers burst into the room, having obtained the warrant needed to search the suspected criminal's hotel lodging.
To burst out is a similar expression to burst into that means to release something that has been under pressure. It is associated with the start of something.
Ex: Though he tried to maintain his composure, the employer finally burst out in laughter at the ridiculous excuses his employee made for the poor job he had done that day.
Ex: If you do not maintain a fair and orderly class, you will find that discipline issues start to burst out amongst even the best students.
Set Light To
To set light to is to ignite or start a fire or set something aflame. This expression is primarily found in British English. In American English the phrase set something on fire is more common.
Ex: Jill was so excited to make use of her brand-new backyard fire pit. It wasn't even dark and she was already heading out to set light to the logs she had carefully arranged earlier that day.
Ex: If we are going to use the bonfire as a signal to attract passing planes, we must charge someone with setting it on fire each night at the same time.
To set the world on fire is an expression that means to do something that causes a major reaction or change in the status quo. If something sets the world on fire, it has great significance.
Ex: Martin Luther King Jr. set the world on fire with his I Have a Dream Speech which called for equality and friendship among all races.
Ex: The sabre-rattling, although taking place in another hemisphere, threatened to set the world on fire unless diplomacy could be restored.