There is no specific moment at which late match starts. You can assume that it starts when there are only a handful of players left standing and they try to remain in their fortresses, whereas storm circle is getting smaller and smaller, within smaller intervals.
In this part of the match, you should have two weapon types – sniper rifle and assault rifle (as good as possible) and this is a must. However, to stand any chances at survival, apart from that you should also have an RPG or a grenade launcher. If you top it up with several meds, you will be ready for fire exchange with the rest of the players.
The first thing to do is build a fortress that will end up in the center of the storm. It is best to build one near the rim of the circle, using metal only. Metal structures are the only ones capable of withstanding a rocket blow. With a wall and stairs on it, your fortress will withstand four blows, before you have to rebuild it. Since you already know to build your fortress near the rim of the circle, also remember to build it on an elevation. Preferably, this should be the highest point on the map. This way, you will not have to bother about attacks from above and you will have a perfect scouting position.
Afterwards, the only thing standing between you and victory is your shooting skills. Later i8n the text, you will learn how to fire to hit your mark in Battle Royale. Regardless of whether you win or lose, the match will end and you will return to the main menu.