The word "stream" has been chosen because it represents (in real life) a very similar meaning to what we want to convey when we use it.
Let's forget about the backing store for a little, and start thinking about the analogy to a water stream. You receive a continuous flow of data, just like water continuously flows in a river. You don't necessarily know where the data is coming from, and most often you don't need to; be it from a file, a socket, or any other source, it doesn't (shouldn't) really matter. This is very similar to receiving a stream of water, whereby you don't need to know where it is coming from; be it from a lake, a fountain, or any other source, it doesn't (shouldn't) really matter.
That said, once you start thinking that you only care about getting the data you need, regardless of where it comes from, the abstractions other people talked about become clearer. You start thinking that you can wrap streams, and your methods will still work perfectly. For example, you could do this:
int ReadInt(StreamReader reader) { return Int32.Parse(reader.ReadLine()); }
// in another method:
Stream fileStream = new FileStream("My Data.dat");
Stream zipStream = new ZipDecompressorStream(fileStream);
Stream decryptedStream = new DecryptionStream(zipStream);
StreamReader reader = new StreamReader(decryptedStream);
int x = ReadInt(reader);
As you see, it becomes very easy to change your input source without changing your processing logic. For example, to read your data from a network socket instead of a file:
Stream stream = new NetworkStream(mySocket);
StreamReader reader = new StreamReader(stream);
int x = ReadInt(reader);
As easy as it can be. And the beauty continues, as you can use any kind of input source, as long as you can build a stream "wrapper" for it. You could even do this:
public class RandomNumbersStreamReader : StreamReader {
private Random random = new Random();
public String ReadLine() { return random.Next().ToString(); }
}
// and to call it:
int x = ReadInt(new RandomNumbersStreamReader());
See? As long as your method doesn't care what the input source is, you can customize your source in various ways. The abstraction allows you to decouple input from processing logic in a very elegant way.
Note that the stream we created ourselves does not have a backing store, but it still serves our purposes perfectly.
So, to summarize, a stream is just a source of input, hiding away (abstracting) another source. As long as you don't break the abstraction, your code will be very flexible.