Progressive overload simply refers to making training harder over time. This can be done in multiple ways, such as adding repetitions, adding weight or adding sets. Though a relatively simple concept overall, it is often misunderstood and mistaken for being the reason in and of itself that progress occurs. This isn't entirely accurate. In reality, it is the act of training within a certain range of your limits (overloading) that causes the body to have an adaptive response to your training and become bigger and stronger. Progressive overload is simply a tool used to keep you within this range as you get stronger. It is not the act of progressive overload in and of itself that drives the progress.
So what does it mean to "overload"?
"Overload" refers to training with an intensity that puts you near your limits in order to force the body to adapt and become better. In the context of resistance training, "overloading" simply refers to training within a certain proximity to failure during a working set. Research shows that training close to failure is what produces an adaptive response in the body, spurring progress. It is typically suggested that you need to be at least four repetitions from failure or closer for this to occur. As long as your training sets bring you within this range of failure, you are creating a powerful enough stimulus to force the body to adapt and spur progress - this is known as "overloading".
How do we apply progressive overload?
As you progress and get stronger, what you were doing during your workouts before will no longer place you close enough to failure to continue making progress. Because of this, we need to progressively overload by doing more so that we can maintain the proper intensity levels as we adapt to our training and become stronger and bigger. This can be done by adding sets, reps or load to get yourself back into the correct intensity zone to stimulate progress again. It's not the progressive overload itself that generates progress, but the act of training close to failure. However, progressive overload is required to keep you within this intensity range. It's basically a tool to chase this intensity zone.
An example of this is making strength gains on a movement such as the bench press. If you train for a period of time doing 3 sets of 10 with a load of 200 lbs, and at the end of this training period the same sets, reps and weight gets you no where near failure because you're now stronger, you will have to use the principle of progressive overload to change a variable in your training so that your bench press sets are close enough to failure again to stimulate progress. You could do this in a variety of ways, such as doing more repetitions with the same load, adding weight to the bar, adding additional sets or any combination of these methods. Adding sets, however, should only be used when the sets you are performing are already at least 4 reps from failure or closer, otherwise you may just be performing a bunch of sets that are no where near failure and never actually achieving overload - often called "junk sets".
Progressive overload requires actually overloading.
Simply adding weights, reps or sets by itself will not guarantee progress, you must make sure that when you're making your workouts harder, that you're doing so by a large enough margin to keep yourself within the proper intensity range that will produce an adaptive stimulus. You must actually be overloading in other words. For example, if in your current training each set you perform leaves 6 reps in reserve, which is shy of the 4 reps from failure threshold mentioned previously required for overload, and you then increase the weight resulting in your RIR falling to 5, you're still not in the correct intensity range that would cause an adequate adaptive stimulus to drive progress. Despite the fact that you made your workout harder, this would not be considered overloading, because your body is not being stressed enough to make gains even after the load increase.
This is why progressive overload needs to be well understood. You should get into the correct intensity zone to drive progress and then apply progressive overload when needed to stay in this range. If you attempt to apply progressive overload and your training is still too easy to drive progress, then you need to make sure you're actually adding enough to your workouts to make it more challenging and drive overload.