"The exploration of logic means the exploration of everything that is subject to law" (6.3): logic determines the form that laws of nature can take, but does not itself make any claims regarding nature. Scientific laws themselves do not belong to logic, because they make claims about experience and do not hold a priori.
The law of induction, the law of causality, and other such scientific principles are not exactly empirical facts, either. Wittgenstein calls them "a priori insights about the forms in which the propositions of science can be cast" (6.34). They define the framework within which we can talk about natural phenomena. At 6.341, Wittgenstein compares the laws of nature to a square mesh laid out over a surface of black and white spots. This mesh allows us describe the surface by saying of each square in the mesh whether it is black or white. Of course, a triangular mesh or a hexagonal mesh could be used just as well as a square mesh, though certain kinds of mesh will likely provide a simpler and more accurate description of the surface than others. And while the mesh itself can tell us nothing about the distribution of black and white on the surface, we can learn about the surface by observing what kinds of mesh describe it most accurately.
The laws of nature cast the form that any description of the world must take. They tell us nothing about the world, though we can infer something about the world from the fact that it is more easily described by one system of mechanics, say, than another.
We need to construct a system of mechanics in order to understand nature. Following Hertz, Wittgenstein claims: "only connections that are subject to law are thinkable" (6.361). That is, we can only make sense of natural phenomena if we see them as conforming to some kind of regularity. We do not actually see causes and effects in the workings of nature, but we can only make sense of the workings of nature if we read cause and effect into them.
The laws of nature do not tell us about nature as much as they tell us how we are going to make sense of nature. They are not necessary truths—only the laws of logic are necessary (6.37)—nor are they explanations of natural phenomena. Wittgenstein compares laws of nature to what the ancients would have called "God" or "Fate": they are the stopping point at which we acknowledge our explanations can go no farther. The mistake of modern science is to see these laws as providing a full explanation of nature rather than just a contingent framework for describing regularities we find in experience.
Just as there is no logical connection joining two events causally, so there is no logical connection between my will and the world (6.373): neither my will nor natural phenomena have any effect on what is impossible and what is necessary.
just to add another insight to a fine summary
have a look at