I wrote a little text that outlines why vector bundles and locally free sheaves are the same thing. This approach is very messy with a lot of gluing, mostly because I did not look at Exercise II.5.18 in Hartshorne right away. The construction given there is much more canonical and preferable over mine. However, I decided to put this online simply because it is different and personally, it gave me a better feeling for why the two notions coincide.