What does Linear mean in Linear Space (Vector Space) The course I'm taking defines a vector space or Linear space as

The vector space $\mathbb{R}^n$ has a linear structure with two features: vector addition and scalar multiplication"

What does it mean by "linear structure" here? (and what makes it linear)

Other people alluded to the formal definition about being closed under linear combinations. Perhaps what you're looking for is a slightly more intuitive, less rigorous answer.
A finite dimensional vector space is isomorphic to some $\mathbf{R}^n$ for some $n$. So for instance, a $1$-dimensional vector space "looks like" the real number line, and a $2$-dimensional vector space "looks like" a plane.
Continuing with this reasoning, I offer the following answer to your question.

A "linear" space is a space which is "flat". So linear essentially means flat in this context.

