The promise communicated to and through Solomon, long-cherished by the covenant people of the Creator God and recorded in the second book of the Chronicles, follows from the Creator God saying to Solomon “When I close up the sky so that it doesn’t rain, or command locusts to devour the land’s vegetation, or send a plague among My people” (7:13). Significantly, this causes the promise to fit into the overall narrative, thus providing a context that is so often lost in much popular use, because when these things are mentioned, Solomon and those to whom these words would be communicated would naturally think of Egypt and the exodus.
Connected to that, (and regardless of when these works reached their final form, as Nicodemus and his contemporaries, like the modern reader, would have been familiar with these texts as part of Israel’s historical narrative) it is likely that such things would also call to mind the Levitical and Deuteronomic curses that were offered up as Israel’s fate, principally for the sin of idolatry. Because of that, these words would ultimately be tied to the idea of exile.
So, making the connection, when the serpents (as recorded in the book of Numbers) attack the people, a people in the process and state of exodus becomes, briefly, a people in exile. Returning to that story then, after Moses prayed for the people, “The Lord said to Moses, ‘Make a poisonous snake and set it on a pole. When anyone who is bitten looks at it, he will live.’ So Moses made a bronze snake and put it on a pole, so that if a snake had bitten someone, when he looked at the bronze snake he lived” (Numbers 21:8-9).
Now, getting back to Nicodemus and Jesus, it would not be unreasonable to think that Jesus expected Nicodemus to have all of these things in mind, and that Nicodemus did, in fact, have such things in mind. Otherwise, why would the author include this exchange in a story that itself would be so thoroughly dependent on the Israel narrative for its own meaning? That considered, one is also reminded that Jesus spoke of believing in Him and having eternal life, just as was the case with Moses and the serpent (eternal life understood as the life of the age to come entering into the present---the idea of living forever in heaven would not here be present).
Those that looked at the serpent did so, presumably, because they then believed their God and believed Moses. Those who looked were healed. They were snatched away from their brief state of exile, experiencing a new exodus. Though many were said to have died, and though, to be sure, many more were going to die if their God did not mercifully intervene, those who looked at the serpent received something akin to a resurrection. Their movement from exile, as they regained their exodus, was a movement from death to life.
This analysis then helps to provide some context for Jesus’ usage of the story. With this reference to the story of the poisonous serpents that were said to be a dramatic memory of Israel’s wilderness experience, Jesus would seem to be informing Nicodemus that the covenant people of Israel, in their own day, were in the same position as the Israel that was suffering from the bite of poisonous serpents. They were not truly trusting their God. They were not believing Him and aligning themselves with His kingdom ways. They were a people in exile, in need of another exodus---a resurrection.