You will notice that besides the two 60 amp breakers, which is called a "double pole breaker", there are other breakers on their side and possible one or two spaces down.
All this group of about six to eight breakers are independent. They mostly control circuits like the AC, the stove, the furnace, the hot water heater, the water pumps, the garbage disposal, and similar.
Under this group of breakers, the rest of breakers are controlled by the double pole 60 amp breaker.
Then, when you cut off the power of the house (the 60 amp double breaker controlling the electrical outlets and lights), the independent breakers will stay On.
If you want no electric power at all in your house, then you must cut off the rest of "independent" breakers as well.