The strategy is to move 20,000 Libertarians to a small state so that they can form a majority government.
The first part is "invasion." The second part is "taking over."
And yet the head of the NH LP says that it's not really about "invasion and taking over."
The first part is "invasion."
Free, adult American citizens purchasing property or renting a residence in a free, American state, and then moving there, is "invasion"? Only by a very wacky definition of that word (according to which everyone "invades" every single city they move to).
The second part is "taking over."
Actually, you have it slightly wrong in your second part above. 20000 libertarians (assuming they get even close to that) will, I reckon, not be enough to form a majority government of New Hampshire.
That said, their goal is to elect people with like-minded views into office in that state. It is hyperbole, but not wholly inaccurate, to call that "taking over". It's what everyone attempts to do when they cast a ballot at the polls.