to see: to take in a person or thing with the eyes, often something momentous one is promised to witness
This is the half of the Greek verb "to see" that the New Testament uses for the future. When a writer wants to say "you will see" or "they will see," this is the form that shows up, so almost all of these uses look ahead to something a person is going to lay eyes on.
Because it so often points forward, it gathers around the great things people are promised they will one day witness: the Son of Man coming in glory, and above all God himself. "The pure in heart will see God" uses this word, and so does the promise that one day God's people will look on his face. That gives the plain act of seeing a weighty, hopeful edge in many verses, even though the word itself just means to perceive with the eyes.
The same word also lends itself to a common turn of speech: "you will see to it," meaning the matter is now your concern and your responsibility. Pilate uses it when he washes his hands of Jesus, and the Roman official Gallio uses it to brush off a dispute. The literal idea of looking at something shades into the sense of dealing with it yourself.