An abc file is just a text file telling what notes are in a tune.

At the site you linked, when you click on a tune, you'll see something like the random one I picked:

X:294
  T:Atholl Braes #1
  D:Silly Wizard, Silly Wizard (1976)
  O:Scotland
  Z:Nigel Gatherer 
  M:4/4
  L:1/8
  K:G
  z2|D3 E G3 B|AGAB AG E2|D3 E G3 B|AGAB G2:|
  AB|c2 ec B2 dB|AGAB AG E2|c2 ec B2 dB|AGAB G2 AB|
  c2 ec B2 dB|AGAB AG E2|D3 E G3 B|AGAB G2|]

If you highlight it (select the text) and copy (Ctl-C) you can paste it into a text editor like Notepad or a word processor like Word. You can then save it to your computer as a text file, using either the common .txt or .abc file extension so you know what it is.

To play the tune, you need an "abc player" and there are a couple of them available, but someone who's used one in the current century would be better able to advise on a good one to use.

If you save the text as a text file from notepad, you can drag the file onto the icon for the ABC2MID converter, which will convert it to a midi, and you should then be able to just click on the .mid file to hear it played (with most common computer setups).

For an introduction to what abc notation is all about, you might try Walshaw: Introduction to ABC In the left sidebar, there's a link to "abc software" (ABC Software) where there are number of programs you can download - once you've figured out what you're interested in doing. You'll probably want something like ABCMID for which you'll have to scroll down a ways.

ABC is just a notation like a page of sheetmusic. Programs to "play the tune" are somewhat specialized, and you will need to get something to translate the notation into noises.

You'll probably have more questions, so don't hesitate to come back and discuss. We have a number of people here who use abc quite a lot. I'm not really one of them, although I do dabble with it occasionally.

John