I think EnergyNumbers's Answer is an excellent one, but could leave some people a bit mystified by what exactly a "direct ray" is and what exactly its relevance is.
The essential idea here is that a Fresnel lens is an imaging machine: it puts a curvature on a low aberration wavefront so that that wavefront converges. Its working depends on there being approximately plane wavefronts at its input (if it is set up to concentrate light by focussing it) and it is intolerant of aberration. Light scattered from clouds has high aberration wavefronts. There is nothing special about "Frensel"; the following applies to all imaging optics. Consider an unentangled photon coming straight from the Sun (see my footnote): its wavefront is almost plane as it reaches the concentrator from the Sun. There may be a small amount of "twinkle", i.e. the wavefront has been distorted a little through interaction of the atmosphere, but the aberration is small. Therefore the focus can be tight: all the way down to the Abbe limit if there is no aberration.
On the other hand, light that reaches us after interaction with clouds has a wavefront that is wildly contorted. Imaging optics simply will not concentrate it, a fact that can be seen in the high sensitivity of a lens's Strehl ratio to aberration. Think of a laser shone at a roughly ground piece of glass: you see speckle. The light is still perfectly "coherent" in the sense of being highly statistically correlated at different points in space and time, but the wavefronts are all over the place and cannot be concentrated by wavefront processing, i.e. by imaging optics.
Now rays are simply a way to describe the normal to a light field's wavefronts - indeed the ability to think in terms of rays depends on the Eikonal approximation to the Maxwellian, photonic description I alluded to above. Another way to think of all of this is through the law of conservation of étendue, or the second law of thermodynamics applied to light. Incoming "direct" solar radiation is made up of parallel rays, perfectly aligned. Its étendue, or entropy is nought or very near thereto (Look up the Wikipedia page on étendue for its definition). Scattered from clouds, the input to a concentrator is made up of rays spread randomly in all directions - this is a high étendue configuration. This puts a severe limit on how much such light can be concentrated: the concentrator simply cannot decrease the étendue: the beamwidth can only be losslessly shrunken by increasing the spread of angles in the ray bundle describing the light.
So, in EnergyNumbers's Answer, a "direct ray" is to be understood as a member of a low étendue, well aligned ensemble.
And, to answer the problem of high étendue light: we build things like photovoltaics which work with unconcentrated light. There is no étendue problem to get around here: photovoltaics simply live with étendue and do not need any concentrator at all (this is not highly related to why photovoltaics are of low efficiency by the way).
Footnote: I only raise the "photon" description here as it helps demystify why we can treat a laser or a highly collimated incoherent beam i.e. light directly from the Sun, in exactly the same way as far as imaging optics are concerned. If the photon description of light in these terms is unwonted to you, see my answer to "How can we interpret polarization and frequency when we are dealing with one single photon?". Essentially, when you look at light one photon at a time, a laser beam and a collimated incoherent beam are the same: lone photons propagate by Maxwell's equations and so their field has wavefronts and interference and all the other things we wontedly associate with coherent light - in Paul Dirac's words, each photon interferes only with itself - so it is like a little coherent beam. At least, this thinking works for unentangled photons.