This has little to do with size change. Your small image is a GIF, and in a GIF there is no partial opacity/transparency: it’s all or nothing, so this doesn’t allow the partially transparent pixels that make smooth edges (a technique known as anti-aliasing). Below, your big image scaled down in Gimp (top) and the scaled image you posted (bottom). The top image has partially transparent pixels:
To keep the anti-aliasing, save your image in a format that support partial transparency (typically, PNG, like your source image).
(white-on-bright-yellow image above...)
If the GIF format is required (but why would it in 2018?), before exporting to GIF you can use Layer → Transparency → Semi-flatten to create anti-aliasing edge pixels over a predetermined color. But of course the picture can only be shown over a background of that color, which somehow defeats the purpose of the transparent background.
PS: If you make the image color-indexed explicitly in Gimp (Image → Mode → Indexed), then Gimp will show you the layers with the GIF “binary” opacity. If you are in full RGB mode, this only appears when the picture is exported to GIF.