Greyscale image exported

When you create a classified image, it no longer contains RGB data and instead each pixel is represented by a class value/number. What you'll need to do is change the symbology of your raster.

Most likely, it is being displayed with the Stretched symbology. This ends up giving you the greyscale because you only have 1 band/value, but if you change that to Unique Values, you can give each class a specific color. You may have to do this manually, unless you have a csv or something with a list of the value/class pairs.