How were film photos enlarged and printed before digital scanning?

Going from negative film to a printed image is a two-step process.

First, the negative is developed — the latent image on the film brought out and then fixed in place. Now you have a piece of translucent film with a negative image on it.

Second, to go to a the final print on paper, you then essentially repeat the process, shining light through the negative to form a focused image on photographic paper — paper treated with light-sensitive chemicals. Then, this paper is itself developed and fixed. That reverses the reversed negative image again, resulting in a positive print.

If your negative is large — as in large-format photography — you may make contact prints, where the negative sits directly on the paper. This results in an image exactly the same size as the original. (See also "contact sheet".)

If your negative is smaller than the desired print (like 35mm film), you use an enlarger: a projector which shines light through the negative and uses a lens to focus the projection on your paper in the darkroom.

This is just the high-level view. The Wikipedia article for "enlarger" provides more detail, including this image:

You can see why, with a setup like this, the work must be in a darkroom. Otherwise the ambient light would also affect the photo paper, resulting in a fogged image.

There's a bit more at How to proceed after having developed the negatives?, and there are many books and articles on the topic which are easily found now that you know what you're looking for.

The level of detail available in a high-quality photograph taken on film is amazing, but generally — just as with digital — the more you enlarge, the worse your results. Going from a 35mm negative to an 8×10" print is an enlargement of about 8½× in each dimension — or 72 times the area. If you have a chance to see an exhibit of 8×10 large-format contact prints, you should take some time to look closely and appreciate. It took a very long time for photographers working in the 35mm format to be taken seriously — this is one of the reasons the internet fascination with "full frame" is kind of funny.