How and why the phrase "quark force increases with distance"? I have seen that phrase "force between quarks increases with distance" at many resources, some even relatively credible (albeit written for general audience). 
What is the reason behind that, when the area law for the confining phase clearly gives the potential energy as increasing linearly with distance and hence the force being a constant?
 A: Perhaps the resources you have seen are confusing "force" with "energy", which is a common misunderstanding that frequently leads to mistakes in terminology.  Energy must be applied to overcome the color force and increase the distance between quarks.  The formation of a new quark/anti-quark pair when the gluon field becomes too stretched may provide an opportunity to confuse the constant strong force field with the increasing potential energy necessary to overcome that field.
Another source of confusion may be a misunderstanding of the color field, thinking it extends throughout space like an electric or magnetic field, rather than being confined to a narrow string-like gluon field between quarks.
For what it's worth, the Wikipedia article on color confinement says that the strong force between a quark pair "acts constantly - regardless of their distance...", citing T. Muta, Foundations of Quantum Chronodynamics (2009), and A. Smilga, Lectures on Quantum Chromodynamics (2001), so they all appear to have stated it correctly.
A: This is the strong interaction which sometimes called strong 'force'. It is one of four forces which 

100 times stronger than electromagnetism, a million times stronger than the weak force interaction and $10^{38}$ times at a distance of femtometer. 

This strong force actually explains how nucleus of atom stick together despite charge of protons. Quantum Chromodynamics QCD is a quantum field theory of the strong interactions between quarks. Gluons are the mediator of the strong force such as photons in the electromagnetic interaction. Gluon comes from word 'glue' since they glue quarks inside hadrons(protons, neutrons, pion, kaon etc.).
Color charged particles such as quarks can not be isolated, gluons always tend to bound them together. This is the phenomenon called color confinement as Ernie mentioned. That's why you can not observe any colored particles directly. Such a quark confinement by gluons makes them colorless. And all particles we observe are color-neutral. So to take quarks apart, you need to give energy however the farther apart, the more energy you give! In particle accelerators you can go beyond this energy then what happens? No they do not fall apart, hadrons generate 'jets' and color charged particles such as quarks, immediately interact with other color charged particles around to create bound states of hadrons which always color-neutral.   
