Nursing covers, simply put are something breastfeeding mums wear to breastfeed in public spaces.
Before the advent of nursing covers, mums would breastfeed in nursing rooms which are rooms set aside for mums to breastfeed. Nursing rooms are typically found in shopping malls. These nursing rooms would typically have cubicles with a chair/couch.
The downside of these nursing rooms is that there are usually just a few cubicles and it may get busy on a weekend. Imagine having to wait for your turn with a hungry baby. Moveover there is the issue of cleaniness: I have personally come across nursing cubicles with questionable hygiene.
Also, having to look for a nursing room is pretty disruptive when you are already settled down and having a meal at a restaurant or when you are in a busy public area.
Having a nursing cover allows you to breastfeed on-the-go and discretely too.