At first glance, the beginning of our Parsha would belong better at the end of the previous Parsha. For the end of Parshas Tzav describes the first seven days of inauguration of the Tabernacle, and the beginning of our Parsha describes the final, eighth day of inauguration, when the Divine presence finally descended into the Tabernacle. 
By breaking to begin a new Parsha in the middle of this story, the Torah appears to be hinting to us that the eighth day, while superficially a mere continuation of the days that preceded it, actually had a totally different character. So the discussion of the eighth day Shemini must begin a new chapter.
Since there are seven days in the week, it follows that the number seven alludes to the cycle of the natural world. Eight, therefore, represents that which is beyond the world, the most sublime spiritual realm which defies any interaction with physicality. Being truly infinite it can have no meaningful relationship with the finite.
And it is this fundamental incompatibility between "seven" and "eight" to which the Torah alludes by placing Shemini in a Parsha of its own. The Torah is teaching us that "eight," that which is infinite and Godly, and "seven," the worldly and the physical, cannot be mixed.
That is to say, that they cannot be mixed by man alone. But God and His commands, of course, are not bound by the paradox of matter and spirit. Thus, when man follows God's command to perform a particular task with a physical object, we witness a most unlikely fusion of opposites: that physical object, whose very nature is to conceal the presence of God, now becomes a pure expression of the infinite Divine Will.
Thus, the 613 mitzvos are, in effect, 613 bridges between "seven" and "eight." Consequently, it is through the observance of these mitzvos that God's presence will become visible within this physical world with the true and complete Redemption-like the eighth day of inauguration, when "the glory of God appeared to all the people."
(Based on Likutei Sichos vol. 17, p. 92ff.)
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