I am learning the moduli stacks of vector bundles and have trouble understanding some definitions. Let $E$ be a rank $n$ vector bundle over the scheme $X$. We denote by $p_i$ the $i$th projection $p_i:X\times X\rightarrow X$ for $i=1,2$. I would like to understand the following statement.

The scheme of isomorphisms $Isom(p_1^*E,p_2^*E)$ is a principal $GL(n)$-bundle over $X\times X$. We define the symmetry groupoid of $E\rightarrow X$ to be $Isom(p_1^*E,p_2^*E) \rightrightarrows X$.

As a trivial example, take $X=Spec(k)$ for a field $k$. Then the vector bundle $E$ is a simply $k$-vector space and $Isom(p_1^*E,p_2^*E) \cong GL(n,k)$. So it makes sense to make a such definition. For a general $X$, the definition above is hard for me to digest. My questions are:

- Why do we need $Isom(p_1^*E,p_2^*E)$, instead of $Isom(E,E)$?
- What is the two maps $Isom(p_1^*E,p_2^*E) \rightrightarrows X$?
- Why is it reasonable to call $Isom(p_1^*E,p_2^*E) \rightrightarrows X$ the symmetry groupoid of $E\rightarrow X$?

I would appreciate it if some experts could help me understanding these things.