The Grassmannian $\text{Gr}(k,n)$ can be described as the quotient $$\text{Gr}(k,n) \cong \text{GL}(k,k) \ \backslash \ \text{Mat}^*_{\mathbb{R}}(k,n) $$
where $\text{Mat}^*_{\mathbb{R}}(k,n)$ is the set of real $k \times n$ matrices of rank $k$. Every point of $\text{Gr}(k,n)$ can be represented by a $k \times n$-matrix $M$. The Plücker coordinates on $\text{Gr}(k,n)$ are all the $k \times k$-minors of $M$.
I don't really understand this construction. I have been reading about how to construst $\text{Gr}(k,n)$ as a projective variety via embedding into $\mathbb{P}\mathbb{R}$ of dimension $n \choose k$ and expressing the $n \choose k$-minors by vanishing of homogeneous polynomials, but I am not sure how to relate that description to the one above.
I also came across a construction of $\text{Gr}(k,n)$ as quotient of $\text{GL}(k,k)$ by some stabilizer, but this seems to be related to its structure as a manifold, not as a variety, as far as I understood. My goal here is to understand another space, which is a subvariety in $\text{Gr}(k,n)$, but I don't really follow this definition of $\text{Gr}(k,n)$ to begin with.
Any help/explanation would be really great!
That should be $\text{Mat}^*_{\mathbb{R}}(k,n)/Gl(k,k)$. You might find it helpful to instead think of this as the quotient $\text{Mat}^*_{\mathbb{R}}(k,n)/\sim$, where $\sim$ is the relation by which $A \sim B$ if $A,B$ have the same row-space.