Please could someone explain the purpose of the brigade??

Brigades are basically your troop setup. If you have 2 of the same type brigades, you get bonuses.

2 Infantry brigade = shield wall bonus = increases DEF
2 Cavalry brigade = charge bonus = greatly increases speed and DMG for a short period 
2 Ranged brigade = barrage bonus = increases ATK speed


Infantry brigades counters ranged
Ranged brigades counters cavalry 
Cavalry brigades counters infantry.

Let me give you an example:

Enemy haves 3 infantry brigades defending the castle. This means, that you should use only cavalry brigades to attack him. This way you will lose less troops, if any. 

Note: there is so much more to take into account when you are attacking or defending. like enemy heroes, research etc..

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When you reach castle level 25 you can use 4 brigades to attack other players castle/resources and to defend your castle. You can only use 3 brigades when attacking monsters.

You also unlock bigger brigade size limit as you level up your castle. At level 25 you each brigade can hold up to 20 000 troops.

Does the 2x brigade bonus only effect campaigns/ monster hunts? 

I don't see the shield walls, cav charge or ranged speedups happening outside of the campaign and monster hunts. Also, can backups appear outside of campaigns/monster hunts or are they limited to the pve environment?

Brigade bonuses should affect everything, including attacking other players etc. Im not sure about backup troops, but I think that those are only for campaign/monster hunts.

Unit rewards should influence everything, including assaulting different players and so forth. Im not certain about reinforcement troops, however I feel that those are just for battle/beast chases.

Warning has been issued  

This may be helpful. Beginners Guide To Brigades