What is the benefit of the "download will begin shortly" page?

It may partially be related to advertisements. Since there is a few seconds of required waiting, users are more likely to read the ads surrounding the link.

This may also help lowering the workload of server if one machines tries to download the same file multiple times at the same time.

One benefit is that it lets you see how many downloads aren't completed -- if you just have a straight download link then only completed downloads will be logged by your web server, and you'll never see the aborted ones.

When you serve a file that is downloaded, browsers keep showing the page that was shown previously.

Having a "Your download will begin shortly" page ensure that everyone see a convenient page.