A photo service is able to copy negatives and create a colored photo.  How is this done?  Can it be done at home?

There are two ways to create a print of a color negative.

Historically, it was by using an enlarger and projecting that photo onto photo paper, and then developing the print in chemicals similar to the way the film was developed. 

But starting about 10-15 years ago, photo services started scanning the film negatives with a high resolution scanner, and then printing it using digital printer technology, much like you might print a photo at home on your inkjet or laser printer.  Of course, their scanning and printing equipment is industrial grade and quality.

So to answer your question, yes, both can be done at home.  Many people have turned spare bathrooms into dark rooms where they can develop film and then either enloarge and develop the prints, or scan the negatives after developed and print with a printer.

There are Youtube videos on how to develop the film at home.  It actually doesn't look too difficult.  And you don't need much space.  Making prints via enlarger would be much more difficult as compared to scanning the prints and then printing them on a printer. 

Yes, it could be done at home. You'll need a scanner. General purpose flatbed scanners will perfectly do but there are also the smaller and more specialized film scanners. Any of those will come with its own  scanning software. You can download your own software if you're not satisfied with it. Some scanning software will automatically convert negatives to positives (correct colored photos) while with others, you will have to convert the scanned images in a graphics software like Photoshop, Corel or Gimp. Personally, I prefer converting through a graphics software. That way, I have more control on how the colors come out.

You can get a negative scanner quite affordably now and then invert and print the photo in true colour on a home printer or commercial one elsewhere.