Regular and routine exercise is a vital component of sustainable health. Especially as you age, your adherence or non-compliance with an exercise regimen will become a decisive factor in determining how gracefully you make your way physically through the decades. Although exercise also provides benefits to the cardiovascular and skeletal systems, the primary benefit of regular training is felt by the muscular system.
Although biochemically complex, the process by which your body develops muscle is not so complex when stated in plain English. Training the body causes microtramua in the muscles--tiny "injuries" that must be repaired. If the intensity of the exercise is sufficient to convince the body that additional muscle is necessary for survival, the body supercompensates by repairing old muscle tissue and creating new tissue.
One of the primary benefits from exercise is the ability to increase muscle recruitment. Operating under the maxim of "use it or lose it," neglect of muscle use results in atrophy to the muscle and the underlying nerve pathways activating that muscle. You might have noticed that the longer you train with weights, the more you can "feel" the muscle working. This is an example of the increased "mind-muscle" connection that becomes possible through regular training.
One of the benefits everyone knows is the concept of training as a way to increase maximal strength levels. The more you use a muscle (and the more efficient you become at recruiting it), the more your potential for force generation rises. Muscle size is not linearly coordinated with muscle strength; you've probably seen relatively small people at the gym lifting large amounts of weight. These people are more efficient at generating maximal strength with the muscle tissue they have.
Another effect of exercise on the muscular system is an increase in muscular size. So long as you continue to progress in the gym by lifting more weights, the body will continue to adapt--adding additional size (and strength) to meet the imposed demands. Your body's primary goal is survival, and by training with heavier and heavier weights, you are telling it that it must grow if it hopes to "survive" the workouts to come.
On top of increased efficiency at recruiting the various muscle fibers (the mind-muscle connection), with consistent training your body will also increase its endurance. This is because the muscle becomes more capable of disposing of accumulated waste (such as lactic acid buildup) even while in the midst of a set. So while during your first week of training you might be able to perform only five reps before "the burn" becomes too intense to continue, you might be able to perform 10 or more reps the following week. This is not necessarily because your muscle became "stronger." It could be because your muscle became more efficient at disposing of waste products through the bloodstream.