You can think of a stream like one of those old-fashioned news ticker tapes. To write data onto the tape, the bytes (or characters) in the file are typed out, one at a time, on the continuous stream of tape.
We can then wind the tape back to the beginning, and start reading it back, character by character, until either we stop or we run off the end of the tape. Or we could give the tape to someone else, and she could do the same. Or we could read, say, 1,000 characters off the tape, and copy them onto another tape which we give to someone to work on, then read the next 1,000, and so on, until we run out of characters.
Once upon a time, we used to store programs and data in exactly this way, on a stream of paper tape with holes punched in it; the basic technology for this was invented in the 19th century. Later, we got magnetic tape, although that was less than useful in machine shops full of electric motors generating magnetic fields, so paper systems (both tape and punched cards) lasted well into the 1980s (when disk systems and other storage technologies became more robust, and much faster).
The concept of a machine that reads data items one at a time, and can step forward or backward through that stream, goes back to the very foundations of modern computing. It is one of those highly resilient metaphors that only really falls down in the face of highly parallelized algorithms: a single input stream is often the choke point for scalability in that case.
To illustrate this, ...