Maybe an example helps. Let $B$ be a constant magnetic field. Then we can take $A=(1/2)B×x$. Now $$(p+qA)^2/(2m)=p²/(2m)+q/(2m)(p⋅A+A⋅p)+q²A²/(2m),$$ and $$p⋅A+A⋅p=l⋅B$$ where $l=x×p$. Thus $$(p+qA)^2/(2m)=p²/(2m)+q/(2m)l⋅B+q²A²/(2m).$$ Here we recognise the $l⋅B$-term as the Zeeman term.
If we now calculate the derivative of $p$ according to $$dp/dt=i[p,H],$$ then we obtain the Lorentz force.
This is the quantum version. In the classical case the commutator is replaced by the Poisson bracket.