Are fields intuitive choices to explain forces? Fields are used to explain all the forces with respective particle being their facilitators. However, are fields intuitive choice to explain the forces? The physical significance is not very apparent. Why were they chosen to explain the forces?
The question is not about the fields being "real" but whether the choice of field to explain force an intuitive choice.
 A: 
Fields are used to explain all the forces with respective particle being their facilitators

In this sentence  disparate concepts are mixed: forces, which are classical level mechanics concepts, and "particles" , which are a quantum mechanical concept, and "facilitators", which is a quantum field theoretical concept.
The definition of a field in physics should be kept in mind:

In physics, a field is a physical quantity that has a value for each point in space and time.

.....

A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor or a tensor, respectively. A field has a unique tensorial character in every point where it is defined: i.e. a field cannot be a scalar field somewhere and a vector field somewhere else. For example, the Newtonian gravitational field is a vector field: specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point. Moreover, within each category (scalar, vector, tensor), a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators respectively

In classical mechanics and electrodynamics, the concept of a field simplifies the mathematical formulations , and particularly for electrodynamics it unified electricity and magnetism, which were thought as separate physical manifestations before Maxwell.
Simplicity of formalism is also the reason for the quantum field theoretical approach, where each particle in the elementary particle table is considered as a field on which creation and annihilation operators generate the physical particles; for all in the table, not only the ones associated with the three forces.  The reason the photon, the gluon and the W/Z are considered carriers of the electromagnetic, strong and weak force respectively  is because the dominant first order Feynman diagram giving the interaction cross section of two particles under the given "force"  have the photon, gluon, W/Z exchange as dominant.
The choice then comes from Occam's  razor or from KISS, i.e. simplicity of formalism for the physics theoretical models.
A: Please note that disturbances of the force-carrying fields themselves travel with finite speed, carrying energy and momentum. Therefore it is quite natural to think of fields not as force carriers introduced in mathematical description but rather as entities as real as particle themselves. Electromagnetic field is just as real as atoms you are built of. It is just another type of matter.
A: Intuition is commonly defined as the ability to understand something instinctively, without the need for conscious reasoning.
So whether or not a concept is "intuitive" is a question of how our minds work, rather than how physics works. Some actions are so well rehearsed that they become "instinctive."  The same is true for our thinking. Intuition is subjective, depending on personal and cultural habits of thinking. In one culture it is instinctive to look for spriritual causes and motives; in another culture it is instinctive to look for motiveless material causes. 
It is common sense to most people that an object set in motion will stop moving after some time. If it keeps going, like a cannonball or arrow, it is "instinctive" to presume some force keeps it in motion. To a person trained for years in Newtonian physics it is obvious that the object will continue moving in a straight line forever. If the object slows down, it is instinctive to presume that a force is opposing the motion. If an object suddenly falls over, it is instinctive for some people to look for an explanation in terms of unseen human intervention, even to the extent of presuming there is a "poltergeist." For a physicist, "intuition" suggests some thermal expansion or mechanical relaxation process is the cause, even without working through the details.  It is "intuitive" to a physicist that any proposed "unlimited free energy device" cannot work, without examining any of the details.
When it comes to explaining "action at a distance" effects such as gravity, electrostatics and magnetism, to someone accustomed to the idea that there must be contact between A and B for one to influence the other, then the assumption that there is some invisible force field around A and B is "intuitive".  This is reinforced by the pattern of iron filings sprinkled onto paper above a magnet : it suggests that something invisible exists in the space around a magnet, even when the iron filings are not there. 

The issue of why fields were chosen is perhaps only explicable from an  examination of the history of science, which would also show what alternative concepts were considered.
A: The field concept is a way of explaining "action at a distance", which was not dealt with in any comprehensive or consistent way since Newton's time. 
For example, Newton's law of gravity  implied that if the Sun vanished, the Earth would feel the effects immediately, as opposed to the real 8 minute delay.
In the early 19th century, the field concept was developed by Michael Faraday, at a time when James Maxwell was beginning to see that both electricity and magnetism could be combined in one set of equations.
From Electromagnetism Wikipedia

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes theelectromagnetic field in space-time of a physical system. The field tensor was first used after the 4-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows some physical laws to be written in a very concise form.

In the early part of the 20th century, the Schrodinger Equation was developed and, although it had no immediate connection with a field theory, the fact that it cannot deal with the creation of particles and is not compatable with Special Relatively, the search for an equation that could cover both of these issues led to:
The Dirac Equation 

Which overcomes both of these problems.  The Dirac Equation led to a reinterpretation of the fields as operators, Second Quantization, with the capability to create and destroy excitations of the field, i.e. quanta.
The position operators in the S.E. are viewed as parameters for the field operators in the Dirac Equation.
The Dirac Equation was not comprehensive enough to cover all particles and forces, but it was a forerunner in the the development of modern day QFT, Quantum Field Theory.
To address your question, is the field concept intuitive, or mathematically convenient,  Sammy Gerbil explains the intuitive aspect better than I could.  A combination of the gradual  acceptance of the idea of the field, accompanied  by its "handyness" in defining and solving more and more problems, inevitably led to its present prominence in the physics toolkit.
A: Force as a concept was introduced by Newton in his laws of motion and his law of gravitation. In his theory the gravitatonal attraction of the sun instantanously affects that of the earth - he found this troubling: what mechanism allowed the influence to travel from the sun to the earth?
Generally, it Faraday thats credited with demonstrating the notion of the field concept in electromagnetism. Sprinkling iron filings on a piece of paper held above a magnet shows 'field lines'; from there, the mathematical notion of a field was developed by Maxwell;and later adopted by Einstein, who showed that the field in gravity was simply spacetime itself.
Mathematically, a field assigns a value to each point in a space; this value can be a scalar, a vector or an operator etc; usually some notion of continuity or smoothness is imposed, so values near the same point are close. 
(The modern way of looking at this, is through the notion of a bundle; so a vector field is a section of a vector bundle - in fact, the sections form a sheaf).
