The problem is you are using wget to download the file.
If you want to stream the file to your users, you have to:

1) Open the stream to the direct link.
2) Send the required headers ( such as the cookie ) | So now you're writing to the stream.
3) Read the response-headers. | Still the same stream, now you just read from it.
4) Send headers to client ( header() function )
5) Echo rest of response to the screen. | You're reading data from the stream in chunks

But note that PHP isn't the right/best language for this.