True/False: $\mathbb{R}^2$ is a subspace of $\mathbb{R}^3$? The answer is False. 

Since $\mathbb{R}^2$ is not a subset of $\mathbb{R}^3$.

However, can someone explicitly define the difference between a subset and subspace? I am somewhat struggling with the technicality of the following definition

Subspace: a subspace, U of a vector space V, is a subset of V, $U \subset V$, such that U is a vector space in its own right, over the same field.

Vector space: A vector space over a field is a set $V$ of objects that is closed under binary operation ('addition') and scalar multiplication

Also, what would make $\mathbb{R}^2$ a subspace of $\mathbb{R}^3$ if anything?

Thanks!
The set of things in $\Bbb{R}^3$ is $3$-component real vectors.  The set of thing in $\Bbb{R}^2$ is $2$-component real vectors.  (The component count is exactly what is being communicated by the "$n$" in $\Bbb{R}^n$.)  Since there are no $2$-component vectors in $\Bbb{R}^3$, no element of $\Bbb{R}^2$ is in $\Bbb{R}^3$.  So $\Bbb{R}^2$ is not a subset, and hence not a subspace, of $\Bbb{R}^3$.
Any set of elements $\in \mathbb R^3$ is by definition a subset of $\mathbb R^3$. A subspace is a particular subset which is a vector space, that is it is closed by multplication by scalars and by addition.

Therefore since we have that for any

$$\vec v\in \mathbb R^2 \implies \vec v \not \in \mathbb R^3 $$

then $\mathbb R^2$ can't be neither a subset nor a subspace of $\mathbb R^3 $.

What is true is that any plane through the origin is a subspace of $\mathbb R^3$.