So the above chart comes from the World Health Association. Basically the two lines represent a range of effect that muscle mass and strength have on a person’s ability to function. The more lean mass you have (the upper line), and the more strength you posses the longer you will be able to perform ADL's (activities of daily living) before you reach the disability threshold.
As we age we loose muscle mass (sarcopenia) which makes it harder to perform basic activities. For example, the chart below shows the relationship between force production and motor units recruited (A motor unit is a motor neuron, located in the spinal cord, and all of the muscle fibers that it innervates). Basically, the more motor units you recruit, the more muscle that is involved. So a 1 rep max deadlift would, theoretically, be at the far right of the 1st graph below (graph A) while walking would be near the left. So the below graph is showing how muscle atrophy effects force production and recruitment. Using their numbers a healthy person requires 25% of their motor pool to activate the muscles necessary to produce enough force to step. Now, someone, say in their 80's that has lost a significant amount of muscle mass has to recruit 40% of their motor pool to produce the same step. Also, the maximum amount of force they can produce is significantly less. So activities, such as walking up the stairs or sitting up from a chair, require significantly more motor units to produce the necessary force. Often times, it becomes a maximal effort for the elderly. Image the effort you have to put forward to do a 1 rep max squat, now imagine that’s what it’s like getting out of your chair.
Returning to the original question, Resistance training helps to maintain muscle mass well into our latter years which in turn makes stepping, walking, sitting up, running, playing, climbing, hiking, opening a jar of pickles, and all other activities much easier.