Jesus got his godly powers and Spirit from God and was sent by God. When God sent the spirit, it is meant to carry some significance (he gives it fully since it's not "by measure"), so when Jesus speaks, he speaks the word of God (3:34). In fact, God does so since he "loves the Son" and "has given all things into His hand" (3:35). From context, it likely refers to the full Spirit Jesus got as well as the highest authority Jesus has, but it's implied to be more than that.
Likely from the importance of Jesus's words, John reiterates what Jesus said -- believing Jesus means eternal life. Additionally, unbelief not only keeps us from "see[ing] life" but also means "wrath of God abid[ing]" on that person (or remaining depending on the translation used). Wrath is a difficult concept, so I'll talk about it in the next paragraph.
The truth is that God's wrath is not devaluing hate on our value -- he loves the world (3:16)! Unlike most human anger, wrath is focused on ensuring that injustice stops and that our lives aren't ruined by bad actions called "sin". While wrath is towards people, it is mainly against crappy actions, much like a parent upset at a kid for scribbling on a wall. From the words of ADW, God's wrath is a "passion to set things right." This is also further supported since the Greek word used for wrath (Strong 3709) conveys a sense of passion. For more references on wrath, Brandon Vaughn from Adventures in Grace talks about the meaning of the Greek word in his article, and Character of God discusses wrath in this article.
In summary, because unbelief means not receiving renewal/clean up, our internal problem remains -- hence God still wanting us to be renewed. At the end of the day, the important part is that because Jesus's words are words of God, we need to pay attention to it -- especially words on how we can be redeemed and get eternal life.