Just to be sure we're working with the same set of definitions:
If you have a set of vectors, their span is the smallest vector space that contains all of them.
A set of vectors is dependent if one of them is already contained in the span of the others. A set of vectors is independent
When only two vectors are concerned, deciding whether or not they're independent is a simple matter. If the two vectors lie on the same line, they are dependent, because either one of them spans the whole line. But if the two vectors do not lie on the same line, then they are independent, and together they span a plane.
When you have three vectors, it gets a little bit more complicated. First of all, notice that in order for all three to be independent, then we surely need the first two vectors to also be independent. So by the above comments, the first two vectors span a plane. Now, in order for all three vectors to be independent, then by definition the third vector must not be in the plane spanned by the first two vectors. So it is outside this plane and the three vectors span a three-dimensional space.
Thus, it is impossible to have three linearly independent vectors in a two-dimensional vector space $V$ -- they need at least three dimensions in order to have the appropriate breathing room. That explains the at most part of the statement.
EDIT: The at least part of the statement follows because if you have only one vector, it only spans a line. You would need an additional vector in order to span the plane $V$.
By the way, linear independence doesn't require orthogonality.