You are "looking into the PAST" the same way you're "hearing the past" when you hear an echo. It's FAR enough away that it takes light about 1.3 seconds to travel from the moon to your eyes. Theoretically, you are seeing what was happening on the moon 1.3 seconds ago which, under normal circumstances means NOTHING. If the moon moved quickly across the sky or if the moon exploded or things happened on the moon's surface that you wanted to watch happen in real-time, then you might care.

Similarly, the sun is far enough away that it takes almost 500 seconds (or 8.3 minutes) for light to arrive. If you assume the light travels a linear path from the sun to your eyes, you actually see the sun set 8.3 minutes after the physical body of the sun passes below the horizon. In other words, if you could (with no other consequences) STOP the sun from "moving" across the sky 8.3 minutes before "sunset" (as perceived from your position on Earth), you would watch the sun finish setting before you realized that it stopped. If you started it moving again, it would take 8.3 more minutes before the light reached you to inform you that it had.

In a way, you can just imagine it all the same way you get LIVE TV- shortly after it actually happens, but close enough to the real thing that it really doesn't matter.