When I download a gif from my website script, it appears like a thin line.
I expected to see the 1" by 1" graphic solid color image that appeared on the site. Is it just a thin line that expands once the web page size is set? How does this work?

It's an old method of creating 'spacer' gifs, by setting the height or width to whatever size 'spacer' you need. Because it's usually 'invisible' and very small, it adds next to nothing to page weight.

So are you saying yes it's just a thin line that expands once the web page size is set?

It's a thin line until you set a width or height for it on the <img> tag that holds it, yes. Spacer gifs should no longer be necessary if you're using CSS, all spacing can be achieved with css.