It depends a lot on your input raster if the output is smaller or larger. Qgis uses the tiff-format as standard, because it is an open format. It also uses no compression as standard for interoperability reasons with other software.
When you input for example an .ecw file you will end up with a file that is often 100 times larger. First of all .ecw is not loss-less, as the standard .tiff is. Second point is that the .ecw is proprietary. When you install Qgis with the support for this type you agree the license agreement and can read this format. But you can´t create one without buying the special software for this. That is the case with many other formats.
You can get a smaller .tiff file when setting in the clip tool in the small field in the lower area the gdalwarp options to a different compression. For example:
gdalwarp -q -cutline cutline_file.shp -crop_to_cutline -tr 1.0 1.0 -of GTiff -co COMPRESS=DEFLATE -co PREDICTOR=2 -co ZLEVEL=9 input.tif output.tif
Different options are possible, like LZW compression or JPEG (smallest file, but lossy). Not all programs can understand all compressions is a point that might be a problem when exporting it to use within other software.
You get only one file in the output as Qgis uses here the Geotiff standard which puts the coordinate reference in the header of the file itself and not in an extra file.