Clipping raster-image increases file size

When you run the raster clip tool in QGIS, it shows you the exact GDAL command that will be run in the bottom section of the dialog box.

By clicking the pen button to the right of the command, you can edit it directly and add in a compression configuration option. In my example, this is the starting command:

gdalwarp -q -cutline /tmp/mask.shp -crop_to_cutline -of GTiff /tmp/453A.tif /tmp/output.tif

You can change this to be:

gdalwarp -q -cutline /tmp/mask.shp -crop_to_cutline -of GTiff -co "COMPRESS=LZW" /tmp/453A.tif /tmp/output.tif

The extra -co "COMPRESS=LZW" is just one of many possible compression options you can use with GeoTIFF files. The GDAL documentation has more information on all the options.