I'm having trouble comprehending an informal proof of the fact that all compact surfaces with boundary can be realized in $\mathbb{R}^3$. I'm trying to find a proof of it on the internet, but I can't find it anywhere. Could someone please give a reference or give a proof of it, starting from the classification of compact surfaces (with or without boundary) ? The proof I'm trying to understand starts from a disk $D$ and glues rectangular strips to change the Euler characteristic and the boundary number (= the number of holes made in $D$), but I don't understand how the gluing happens.
Any help is as always appreciated.