Ammonia for cleaning - what is the interaction between ammonia and oils?
Common food fats and oils usually contain some free fatty acid1. The alkalinity of household ammonia serves to form a soap which will emulsify the main body of the fat/oil, if enough elbow grease (i.e. vigorous agitation with a scrub brush or wash rag) is applied. Other alkalis do as well - better, if you consider breathable ambient air to be desirable.
Some brands of commercial household ammonia already contain fatty acid soaps and exhibit a cloudy appearance.
It also seems to me that a dilute solution of washing soda ($\ce{Na2CO3}$) would clean quite well, and ammonia should do no better. But when the odor of $\ce{NH3}$ disappears, and the oily/fatty surface is now clean, it is as if there are now two indicators of cleanliness: no odor and no oiliness, therefore the surface must certainly be clean, because all evidence of the cleaning process has been removed. It must be a psychological thing, like foaming with detergents.
Reference:

Analytical approaches for the assessment of free fatty acids in oils and fats
S. A. Mahesar, S. T. H. Sherazi, Abdul Rauf Khaskheli, Aftab A. Kandhro, Siraj uddina, Anal. Methods, 2014 ,6, 4956-4963, DOI: 10.1039/C4AY00344F