Russell's idea in "on the nature of truth" (1906) seems to be, to escape the Liar paradox through the rather colorful suggestion, that there is no such thing as a false belief. In ordinary language we do recognize, of course, false beliefs. Russell's suggestion is that on reality, "behind" ordinary language, there are no false beliefs. What we ordinarily call false belief is, in reality, a belief in nothing.
In the event of the objects of the ideas standing in the corresponding relation, we shall say that the belief is true, or that it is belief in a fact. In the event of the objects not standing in the corresponding relation, there will be no objective complex corresponding to the belief, and the belief is belief in nothing, though it is not "thinking of nothing", because it is thinking of the objects of the ideas which constitute the belief.
What Russell meant by a "belief in nothing" seem to be this: Consider the belief that the sun is shining (at some time and place). If the belief is true, there is a corresponding something in the world (something that Russell called "a fact") that connects together "the sun" and "shining". It is the fact that the sun is shining. But if the belief that the sun is shining is (what we would ordinarily call) false, there is no corresponding fact in the world. If there is no corresponding entity in the world, a fact, then what do I believe in? How can a false belief have an object, when no corresponding object exist? Russell's suggestion was that, for a false belief has as object the ingredients of the intended fact, but not the fact itself. For example, if I believe that the sun is shining, and it isn't, then my belief has as objects the sun and (the universal property) shininess. The belief will have no fact as object, and to that extent it will be "a belief in nothing". Also, since the belief itself will be different when it is true than when it is false, the belief will not be "a single thing" as Russell puts it.
How does all this help with the Liar paradox? My understanding is that it does that by eliminating the concept of a false (or mistaken) belief. It still doesn't seem to me to work, because it merely replaces the concept of false belief with that of empty belief ("a belief in nothing"). So you can replace, in the formulation of the paradox, each "mistaken belief" with "empty belief" and still you end in a similar paradox. In addition, even if this solution worked, the cost would be high. Because it warps the concept of belief in a strange and unnatural way.
I suppose that Russell realized quite quickly the shortcomings of this attempted solution to the Liar. In Russell's later views, the problem of non-existing objects of beliefs was solved by the theory of descriptions. And the Liar paradox he attempted to solve through the logic of types.