Non-commuting matrix exponentials

The equality is true when $A,B$ are quasi-commutative, that is, when $[A,B]$ commutes with both $A$ and $B$.

The author probably means an approximation rather than a true equality, as $(A+B)+\frac12[A,B]$ are only the first two terms in the BCH formula Zassenhaus formula.

The book you referenced was "Quantum Physics in One Dimension". The operators in quantum mechanics have to obey particular commutation relations. For instance the operators $x$ and $p$ obey the canonical commutation relation,

$$ \left[ x, p \right] = i I. $$

This allows us to simplify the Zassenhaus formula. The operator $[x,p]$ commutes with every operator because it is proportional to the identity; if you consider the higher order terms in the Zassenhaus formula you will see they involve commutators which must be zero.

Meaning that,

$$\exp(x+p)= \exp(x)\exp(p)\exp(-[x,p]/2),$$

is an exact relationship for the operators $x$ and $p$.

Knowledge of specific commutation relations can allow you to simplify the Zassenhaus formula significantly.