Branching is a method of giving a deeper more fundamental structure and meaning into sentences by creating branches according to a set of rules.
Lets start with the English sentence He ate a red apple on the sofa.
S / \ / \ NP VP | | [He] [ate a red apple on the sofa]
A sentence can, at its highest point, be divided into a noun phrase and a verb phrase.  The noun phrase contains the subject while the verb phrase contains the verb and the object.
The verb phrase can branch out even further. Because he contains no additional branching, only apple's branching will be shown here.
The VP is divided into a V, an NP and a PP
VP /|\ / | \ / | \ / | \ / | \ / | \ / | \ / | \ | | | [ate][a red apple][on the sofa]
NP / | \ / | \ / | \ Det Adj N | | | [a][red][apple]
PP is divided into a P and a NP
PP / \ P NP | | [on][the sofa]
All together it is like this:
VP / | \ | \____ NP \ / \ PP Det N / \ P NP | / \ On Det N | | the sofa