What exactly is entropy?
Entropy has two general precise definitions.  That which is most useful here is from Boltzmann, S(E) = k ln W(E) where S is the entropy, E is the energy of an isolated system (which is the easiest system to consider), k is Boltzmann's constant, and W(E) is the number of states of the system that have energy E.

For example, if the system has exactly one state with energy E, then S = 0.  If there are 2 states with energy E, then S = k ln 2.  And so on.

Now "organization" has no precise definition, but is rather an descriptive label we might apply to categorize states of the system.  Some of those states look "organized" to us, some look "disorganized."  For example, consider a model of atoms adsorbed on a crystal surface.  Let's model the surface as a square 3x3 array of sites and put 5 atoms on it.  A priori, there are 9!/(9-5)!4! = 126 ways of putting 5 atoms on 9 sites.  Now which of these states are "organized?"  That's to some extent a matter of opinion, but a common definition is "even" or "symmetrical."  So, for example, the state where there's an atom on every other site, in a checkerboard pattern, looks "organized" -- as if the sites formed a 2D crystal.  There's only one such state.

So now let us supposed we observe  this system, and we have some way of knowing, either from measurement or method of preparation, that it is in an "organized" (symmetric, checkerboard-like) or "unorganized" state.  There is only one "organized" state, so if we know the system is in an organized state then its entropy is k ln 1 = 0.  On the other hand, if we know the system is in an "unorganized" state there are 125 possible states it could be in, so its entropy is k ln 125, higher.

This is generally the case.  States we are willing to call "organized" generally have severe restrictions on the degrees of freedom: their values are distributed symmetrically in some way, e.g. the atoms are distributed uniformly, or in a regular array, the velocities are all the same, and so forth.  Because of these restrictions, it is almost always the case that there are fewer states of the system that satisfy the restrictions than that don't.  So the entropy of the system when it's known to be "organized" is less than when it is known to be not.

In short, it might help if when you hear "organized" you mentally translate that to "has some symmetry restrictions on how the degrees of freedom (e.g. atoms) are arranged."  Hopefully it is intuitive that there are far fewer ways of arranging atoms symmetrically than not, and therefore that the entropy of systems known to be symmetric is lower than the entropy of systems that are not.