Let $U = [u_1, u_2,\dots ,u_n] \in \mathbb{R}^{m\times n}$ with $u_i \in \mathbb{R}^m$ and $V = [v_1, v_2,\dots ,v_n] \in \mathbb{R}^{p\times n}$ with $v_i \in \mathbb{R}^p$. Then $$UV^T = \sum_{i=1}^n u_iv_i^T \in \mathbb{R}^{m\times n} $$
I am trying to get more intuitive understanding of matrix multiplications (say thinking in terms of linear mapping) and for the above I know how to prove it using 'tedious' algebra but I am looking for more insightful proof that will illuminate the essence of the above identity.
Think of it like this. You are constructing a matrix $A\in\mathbb{R}^{m\times n}$ via the product $A = UV^T$. $A$ is a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$, but by using this product, you are sort of taking a detour through $\mathbb{R}^p$. The result of multiplying $UV^T$ times a vector $w$ will be a linear combination of the $p$ columns of $U$ (so the range of $A$ has dimension at most $p$) and the mapping from $w$ to which linear combination we take is controlled by $V^T$.