Here it is stated that
These fibers can be glued together in a natural way so as to obtain a principal $GL(n,R)$-bundle over $M$.
I don't understand what they mean by glued together. Aren't the fibers glued already because each is attached to the base space $M$?
The vector bundle is trivialized by/on some cover of your manifold, and of course there are transition maps when you represent the "same" vector that is sitting on top of two different particular open sets, which of course we know. This is what they mean by gluing (of the original vector bundle). Then, the same transition maps "naturally" induce a gluing of the associated principle bundle (instead of thinking of the data vector by vector, we get transition maps by taking a "matrix of bases" to another matrix of bases.)