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Given a vector $c\textbf{v}$ (where $c$ is a scalar) which lies along a line, if a vector $\textbf{w}$ is not one that line, the combinations $c\textbf{v} + d\textbf{w}$ fill the whole two-dimensional plane.

What does it mean that it fills the whole 2-D plane? I can't see the geometric intuition here. The most I understand from this is that I have one point on the real line representing by $c\textbf{w}$. Now if I have another point $d\text{w}$ not on that line I can draw a line between the two points. I also understand that I can choose any two such points in 2-D space. Is that what is meant by filling the whole 2-D plane?

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    Think about basis vectors. All vectors in 2d space are linear combinations of $ \mathrm {i,j} $ ,and if we replace them with two other vectors that are not on the same line,linearly independent,then the situation would not change... – 2017-01-02

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The more usual statement is that given two vectors $\textbf{v, w}$ that are not along the same line, the combinations $c\textbf{v}+d\textbf{w}$ fill the plane. The point is that for any vector $\textbf x$ in the plane you can find $c,d$ so that $\textbf x=c\textbf{v}+d\textbf{w}$, so the whole plane can be reached by this approach.

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    So my last sentence in my post was loosely correct. – 2017-01-02
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    It is not about drawing a line between points, it is being able to represent any point in this way. The simplest case comes when the vectors are $(1,0)$ and $(0,1)$, where you just choose $c$ as the $x$ coordinate and $d$ as the second. The import of the statement is that for any two (non-colinear) vectors, you can represent any point this way. – 2017-01-02