In both sonnets the topic is love and the faithfulness of the speaker to his love.
Sonnet 116--Love does not alter when if finds a reason to change. It is ever-fixed like the star the boat uses to navigate the sea. Though lips and cheeks may fade with age, love does not alter because the appearance of the loved one changes. It lasts until death. If this is wrong and can be proved, I (Shakespeare) never wrote and no man ever truly loved.
Sonnet 43--I see best when my eyes are closed. All day I am forced to look on what I do not care about; only at night, when my dreaming eyes see you, do I truly see. For even your image (ie, shadow) brightens all the shadows of a dream, and how much brighter you are in daylight, when you may be seen in reality? How much, then, would it delight me to see you in reality, when already your image in my dreams makes them so bright? Until I can see you again, my days are as dark as night because of your absence, and my nights as bright as day because of your sight in my dreams.