Why, in older movies, does the image suddenly appear stretched when end credits run over the footage?

Because the versions you're watching have been converted to display on screens that aren't as wide as the movie screens they were originally filmed for.

This is usually done by cropping less important parts of the image to make it better fit the aspect ratio of a TV, monitor etc. They can't do this for the credits, since it would cut off parts of the words.

That happens on films that were converted to video to show on TVs, such as from a DVD. The height and width dimensions of films meant to be shown in a movie theater are different from the dimensions of a home television. For most of the movie, the technicians converting it for home showing are able to shift the image to the most important part of that scene. For credits, which typically stretch across the screen, they more or less have to convert them directly, so as not to lose any of the words. You can play around with the effect to some extent by hitting the key on your remote to change the aspect ratio of what you are watching.