One day after the priests asked John about the baptism thingy, John actually got to see Jesus -- in person. That person is who John was referring to as the One. In fact, John technically didn't know that the One was Jesus, but John knew that whoever the "One" is, that person is very important.
In fact, John had another reason to know that Jesus would be a very important person. He saw the Spirit enter Jesus like a dove. John knew earlier to look out for someone that is the recipient of a descending Spirit (1:33), and Jesus was the one.
And with his new sightings, John decided to, you guessed it, proclaim that he has "seen and testified" the Son of God (1:34). All these sights are pretty important for others to know! No wonder this section was in the Bible.