For a space $X$, isomorphism classes of rank-$k$ vector bundles on the suspension $SX$ are in one-one correspondence with homotopy classes of maps from $X$ to $GL_k(F)$, where $F$ is ${\mathbb R}$ or ${\mathbb C}$. You get $K_1(X)$ by replacing $GL_k(F)$ with the direct limit $GL(F)$. The algebraic analogue, then, is that $K_1(R)$ should be generated by starting with "homotopy classes" of maps from $Spec(R)$ to $GL_k$, or equivalently elements of $GL_k(Spec(R)$, and then taking a direct limit. So you want elements of $GL(R)$ modulo "homotopy".

There's an obvious sense in which elementary matrices are homotopic to the identity (given a matrix of the form $I+M$ where $I$ is the identity and MM has one non-zero element, consider $I+tM$), so it's natural to try modding out by these. The Karoubi-Villamayor construction is a sort-of-generalization to higher $K_i$ but it only works well when $R$ is regular.

For a space $X$, isomorphism classes of rank-$k$ vector bundles on the suspension $SX$ are in one-one correspondence with homotopy classes of maps from $X$ to $GL_k(F)$, where $F$ is ${\mathbb R}$ or ${\mathbb C}$. You get $K_1(X)$ by replacing $GL_k(F)$ with the direct limit $GL(F)$. The algebraic analogue, then, is that $K_1(R)$ should be generated by starting with "homotopy classes" of maps from $Spec(R)$ to $GL_k$, or equivalently elements of $GL_k(R)$, and then taking a direct limit. So you want elements of $GL(R)$ modulo "homotopy".

There's an obvious sense in which elementary matrices are homotopic to the identity (given a matrix of the form $I+M$ where $I$ is the identity and $M$ has one non-zero element, consider $I+tM$), so it's natural to try modding out by these. 

The Karoubi-Villamayor construction is a sort-of-generalization to higher $K_i$ but it only works well when $R$ is regular.