You should look at streams as abstractions on underlying 'sources' or 'sinks' of data. A source is something you read data from, and a sink is something you write data to.
The concept of streams allows you to perform I/O on various forms of media, network connections, pipes between applications, files, etc.
The stream abstraction is very valuable to us as developers as it allows us to simplify input and output, and it gives us the flexibility to arrange and reconnect the sources and destinations of these streams.
A good analogy is that of a hose. You can send and receive data through hoses, and you can connect these hoses to various things.
By allowing programs to talk through hoses, we allow all sorts of programs to talk to each other, and we increase interoperability and utility vastly.
This is at the heart of the UNIX philosophy, and supports some very powerful programming idioms.