I'm confused about what is being asked here. We're given that if $A$ is an $R$-module, and for each $i$ there is an $R$-linear map $\beta_i : M_i \to A$, then $A$ is isomorphic to the direct sum of $\{M_i\}$.
We were learning about tensor products and its universal property, our teacher mentioned in passing that we can prove this using the universal property, and I was wondering about how one would go about doing so. Thank you for your help. Please don't downvote this for being ill-posed, I'm just following the words that I heard in class, and realize there might be other important information that is missing here that we need to prove this.
I think you missunderstood what he meant, you cannot use the universal property of tensor product to prove such a statement. You can however use the universal statement for sums, aka coproduct, which is that if $M_i$ is a collection of object and $M$ their sum, and there exists a homomorphism $f_i:M_i\to X$, for each $i$, then there exist a homomorphism $\varphi:M\to X$. such that $\varphi\circ\imath_i=f_i$
This is unique up to isomorphism and can be shown that.