How is it different from a classical field ?
For starters, a quantum field is operator valued rather than scalar, vector, tensor, etc. valued. A quantum field assigns an operator to each event in spacetime.
How do we picture a particle arising out of an excitation of its
respective quantum field ?
We have to be careful to distinguish the quantum (operator valued) field from the entity that is operated on and its quanta. See, for example, this answer.
Also, from the first chapter of "Student Friendly Quantum Field Theory", section 1.8:
When the word “field” is used classically, it refers to an entity,
like fluid wave amplitude, E, or B, that is spread out in space, i.e.,
has different values at different places.
By that definition, the wave function of ordinary QM, or even the
particle state in QFT, is a field. But, it is important to realize
that in quantum terminology, the word “field” means an operator field,
which creates and destroys particle states.
States (= particles = wave functions = kets) are not considered fields
in that context.
Honestly, I do not know how to picture or even if it is possible to picture a quantum field and the quanta they create and destroy.
I will be very interested in other's answers as I too have long struggled to find a satisfying picture.