I've always been in the "I don't get it" boat with DDG bang operators. When I last tried switching it to my default search engine, I felt like any time I needed to search DDG for "!{bang} {query}" I could have just searched Google for "{query}" and gotten the result I was looking for. In many cases, Google will give even better results for siteX than siteX's own search, because Google is likely to correct for things like common spelling errors or word substitutions.
Granted, I haven't given DDG an earnest try in a couple of years, so the DDG non-bang search results may have gotten better, but the fact that their bangs are still heavily touted as a killer feature doesn't give me much faith.
I would call my use of bangs getting to a specific page on another site rather that searching another site. For instance, I get a lot of use out of the wolfram alpha bang, !wa, in those uses I'm not searching as much as I'm inputting a calculation for wolfram alpha to perform and the bang let's me skip a step by typing "!wa {quety/calculation}" into my address bar. Similarly, if ddg is your default search engine, you can lookup a url on archive.is just by prepending it with "!archiveis " in the address bar.
On Google I just search for "define wordtodefine" (which isn't must longer and might even be faster to type) or even just "wordtodefine" (Google seems to be pretty good at knowing whether a single-word search is likely to be looking for the definition).
The most obvious reason why you'd use DDG is for the privacy aspects and the bang operators make it super easy to quickly use Google for those times you need it which limits the amount of information they have.
Granted, I haven't given DDG an earnest try in a couple of years, so the DDG non-bang search results may have gotten better, but the fact that their bangs are still heavily touted as a killer feature doesn't give me much faith.