While the emphasis, or main focus, for walking a dog might be on "answering the call of nature," the emphasis for the dog might be on chasing squirrels.
If you say emphasis a little louder or with extra oomph in your pronunciation, you are putting emphasis on the word. This emphasis makes it clear that special attention, or emphasis, should be given to the word. Now, having repeated emphasis so many times, the "s" sound really stands out, sort of like in the word "stress." When you stress a point, you put emphasis on something — it is the main thing, the emphasis.