Consider two adjacent regions A,B,(of water, for example) which has different concentrations of a solvent (for example, salt).
Let the boundary between these two regions be permeable for the solvent.
(Think of A and B as the two halves of a tank)
Now, every molecule of the solvent moves in a random fashion.
Hence, the fraction of molecules in A that passes over into B (that is, the percentege of the molecules contained in A) should equal the percentage of molecules originally in B that passes into A.
But then it follows that the actual number of molecules passing from the region of higher concentration must be higher than the number of molecules passing over from the region of lower concentration.
Hence, you have a net drift of molecules from the region of higher concentration into the region of lower concentration.
It is important to realize that the molecules themselves couldn't care less about which region they belong to; diffusion is the net effect of random motion.