To bounce back is used idiomatically to mean to return to a good, successful or healthy state after a problem. In this case it means to return to a good state of health. Depending on the context this could also involve being slim and good looking.
The preposition back nearly always indicates returning to a previous state or previous location. This can help you guess the meaning of [verb] plus [back] combinations.
The word right is a specialised word (most grammars think it is an adverb). It is often used to modify prepositions. It can have several meanings. It can mean immediately, completely, directly or exactly. This adverb cannot be used to modify adjectives or adverbs in standard English.
In the Original Poster's example the word right in the phrase bounced right back means that her body recovered quickly and without any problems. It also indicates a sense of completeness indicating that her body was just as healthy as it was before she had the children.
Another specialised adverb for modifying prepositions is the adverb straight. It has a similar meaning to right:
- It went straight past me.
- It went right past me.