tl;dr: Steering feedback means different things to different people. Ultimately, however, it's about using the steering wheel as both an input and an output device.
At a very high level, the concept of steering feedback has to do with the user interface of the car. If we restrict the discussion to just the wheel in front of you, it's easy to imagine that that is purely an input device much like the keyboard and mouse in front of me. There's a very obvious example of where that is all it is: video games. Plenty of arcade games in the 80s had wheels that were free-spinning without even the concept of a center point. This is similar to the Hyundai in your example: you feel like your control inputs are only loosely connected to reality and you're certainly not getting any information back.
Once we start thinking about the wheel as a possible output (i.e., feedback) device as well, it helps to think about the system. Remember that your rotation of the wheel is directly connected to the twisting motion of the steered wheels (almost always in the front). Try turning the wheel of a motionless non-power steering car (or one with the engine off). You'll gain a lot of intuition about the way the system works: very hard to turn in a slow high-grip situation vs. much easier to turn in high speed or low-grip situations. So, at a basic level, dynamic steering effort is the most obvious form of feedback.
Of course, there are other mechanical signals that the wheel can send you. You gave the example of rough roads: induced wheel motion is another form of feedback. In some cars, you'll feel every bump and dip in the road through the wheel. In others, it seems like the wheel is floating above the bumpy road. The first can be exhausting if the wheel tries to jump out of your hand. The second can lead to overconfidence and thinking "this road isn't so bad."
So, why do we care? Fundamentally, drivers generally care about transitions and unexpected state changes. For example, imagine an entrance ramp to a highway that you enjoy scampering briskly up. On a dry sunny day, you expect that you'll dial in the steering input you want and feel the car's suspension take a set. Then, holding the wheel against its tendency to straighten out and you'll zoom around the corner.
Imagine, however, it's a day like today (where I live) where it's cold, drippy and there are patches of ice hiding under thin puddles. If you tried the same maneuver, you might dial in your steering and start the turn, again holding the wheel against its tendency to straighten. If your front wheels hit one of those ice patches, however, it would suddenly become much easier to turn the wheel. Hopefully, you'd be in and out of that ice patch before a real low traction scenario developed (probably understeering right off the road in this case) but it does illustrate one of the most useful aspects of steering feedback: an alert driver can take the feedback as a warning about changing or dangerous conditions before they actually become a problem.