This is too simple to be a ‘real’ answer, but I feel like it might nonetheless be worth just having it written somewhere: the best way to think about representations, especially high-dimensional representations, is in terms of linear actions; it is entirely possible that a group can act on one vector space in one way, and on another vector space in an entirely different way, and this should not surprise us any more than that one can interpret $4$ as the area of a square when using $\sqrt4$ to compute the sidelength, or as arclength around a circle of radius $1$ when computing $\sin(4)$. This point of view @joriki has explained very well (and in much better language than my overly prosaic analogy).
However, it is also possible to think about representations in terms of matrices, and it appears that you feel most comfortable at this point doing so. In this connection, you can literally make 8-square matrices out of 3-square ones by stacking them:
$$
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix} \mapsto \begin{pmatrix}
1 \\
& 1 \\
&& a & b & c \\
&& d & e & f \\
&& g & h & i \\
&&&&& a & b & c \\
&&&&& d & e & f \\
&&&&& g & h & i
\end{pmatrix},
$$
or, if you really prefer all the $0$s,
$$
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix} \mapsto \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & a & b & c & 0 & 0 & 0 \\
0 & 0 & d & e & f & 0 & 0 & 0 \\
0 & 0 & g & h & i & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & a & b & c \\
0 & 0 & 0 & 0 & 0 & d & e & f \\
0 & 0 & 0 & 0 & 0 & g & h & i
\end{pmatrix}.
$$
As @KyleMiller said, the resulting 8-square matrix is often referred to as the direct sum of the obvious two 1-square matrices and two 3-square matrices, and denoted, to save space, by
$$
\begin{pmatrix} 1 \end{pmatrix} \oplus \begin{pmatrix} 1 \end{pmatrix} \oplus \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix} \oplus \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}.
$$
The similarity to the direct-sum terminology and notation for vector spaces is not accidental; the action of this matrix on the vector space $\mathbb R^8$ comes from viewing that space as $\mathbb R^1 \oplus \mathbb R^1 \oplus \mathbb R^3 \oplus \mathbb R^3$, and acting as indicated on each summand.
I hope it is clear both that we have produced an 8-square matrix, and how we have done so. Now, of course, the question is “why would we do such a thing?”—and for that, again, it is probably best to defer to @joriki's explanation.
On further reflection, one more thing: I think that also the terminology of ‘representation’ may be misleading here. It is natural to think that a representation of a group is supposed in some way to faithfully represent the elements of that group, but it need not do so—in fact, we have the terminology ‘faithful representation’ precisely to describe the situation where it does—or, in addition to or instead of faithfully representing group elements, that it should in some way be an efficient or natural representation of those elements. This was indeed some of the motivation for the formation of the theory, but, as is usual as theories develop, by this point we have abstracted away from the idea of how good, bad, efficient, or faithful the representation is, and now demand of a representation only that it be a linear action of whatever sort on whatever space; or, if we think on the level of matrices, that it be a homomorphism to an appropriate matrix group, here $\operatorname{GL}(8, \mathbb R)$. Thus, to say that we have an 8-dimensional representation of $\operatorname{SO}(3, \mathbb R)$ is to say literally nothing more than that we have a homomorphism $\operatorname{SO}(3, \mathbb R) \to \operatorname{GL}(8, \mathbb R)$. I have written one such, but there are others; for example, we could send every orthogonal matrix to the identity 8-square matrix.