What is actually a 'Field' in Physics? How can something affect something at a distance? Every text just describe fields mathematically and as a 'vector field' in which it is said a particle gives rise to a field because each point in space around it becomes associated with with a force vector. But it never explains how does a particle generate a field at the first instance, or what a field really is or how does a field work - how is a particle able to affect another one at a distance?
 A: Here is a definition of a field in physics:

In physics, a field is a physical quantity, typically a number or tensor, that has a value for each point in space and time

....

As another example, an electric field can be thought of as a "condition in space" emanating from an electric charge and extending throughout the whole of space. When a test electric charge is placed in this electric field, the particle accelerates due to a force. Physicists have found the notion of a field to be of such practical utility for the analysis of forces that they have come to think of a force as due to a field

Note this:  that physicists   have come to think of a force as due to a field.
That is the crux of the confusion. What can be seen and measured physically is the force. The field in physics is part of a mathematical model, 
 a mathematical representation in space,( scalar , vector or tensor generally,) a mathematical model which is proven to work, i.e. fit existing measurements and predict the value of new measurements in different boundary conditions. In this example predict the force on a test particle.
Since ancient times people tended to think that mathematics represents the underlying reality, and measurements are due to nature obeying the mathematics ( platonic view in a sense).
Physics has progressed now, and it is well understood that physics theoretical models have validity for special boundary conditions:
1)Newtonian mechanics and classical electromagnetism for dimension commensurate with human sizes.
2)Quantum mechanics and quantum electrodynamics for dimensions commensurate to h_bar
3) Special relativity and general relativity for very large speeds and large masses respectively.
These models blend smoothly in areas of overlap, but having them clear in the classification allows to see that there is no unique way of mathematically describing data so that valid predictions can be made. There are just more convenient models. example: one does not use general relativity to calculate the throw of a ball. Newtonian mechanics is quite adequate within the errors of measurement.
In this question of yours 

But it never explains how does a particle generate a field at the first instance, or what a field really is or how does a field work - how is a particle able to affect another one at a distance?

The mathematics and the physics are confused.
The particle exists, its effect on other particles  can be modeled within classical electricity, if it is a charged ball, using the equations for electric fields and the solutions to the problem will fit the data perfectly. In this, one assumption/axiom is "action at a distance" , and there is no problem for laboratory dimensions . The solutions describe within errors all interactions of macroscopic objects.
If it is an elementary particle, an electron for example, it still  can be modeled with the electric field BUT an electron is a quantum mechanical entity, and the quantum electrodynamics (QED) mathematical model has to be used to describe a physical state. In QED there is no action at a distance, all transfers of energy and momentum are limited by the velocity of light, and a complicated formalism exists to predict the behavior of the electrons in specific boundary conditions, at least a semester course. So, no action at a distance at the underlying micro level of quantum mechanics.
As macroscopic theoretical descriptions can be shown to emerge from the microscopic quantum mechanical ones, there is no action at a distance for the classical solutions , but the velocity of light is so huge, the one can ignore it at the macroscopic level.
Similar arguments hold for gravitational fields, which are also dependent on the velocity of light, and this can be seen in the cosmological models of the universe.
In studying physics, one has to keep in mind that the theories describe observations and a successful theory is one that is predictive of new measurements. The concept of a "Field" is a useful variable in the mathematical descriptions for the experimental situation studied, and the appropriate frame has to be chosen to have a simple solution to a given problem.
A: A field is a quantity assigned to each point in space. The quantity can be a scalar (number), a vector, or something more complicated.
A particle doesn't "immediately create" the entire field. It only creates it locally and then the change propagates at a finite speed (typically speed of light $c$). 
The spread of the "update" is also local - changes of the field affects only the (infinitesimally close) nearby values of field. This is described by partial differential equations, usually called field equations or wave equations (because of the wave-like behaviour).
Examples of such partial differential equations are Maxwell's equations (for electromagnetism) and Einstein field equations (for gravitation).
So, at least in the classical physics, everything affects everything else only locally.

EDIT: For a nice explanation of this, there is a nice MinutePhysics video.
A: A field is a mathematical notion (as anna's very good answer explains). What exists is vacuum fluctuations - virtual particles that "connect" real ones to the proper extent. They give rise to the rest. 
