You have to go back and read Numbers 21 to get the context. The Israelites are complaining about their free food in the wilderness. God sends serpents among them to kill them. Then when Moses prays, he commands Moses to set up a serpent on a pole. Whenever someone bitten by a serpent looks at the pole, they live. God therefore functions as both the one who kills and makes alive, who sends the snake that poisons and the snake that kills.
Possibly the most surprising part of the Numbers event is that God uses the same thing--a serpent--to kill and make alive. Who wants more of what already ails them?
This allegorical interpretation that Jesus employs should help us understand "for God so loved the world that He gave His only Son" in a new light. But it will help to go by way of an extended quote of Luther to get the full drift (from WA 5:195. 41):
Whoever would be righteous must first become a sinner; whoever wants to be well, good, and like God as a Christ-like member of the church must first become sick, bad, perverted, devilish, even heretical... as Paul says: 'Whoever among you would be wise must first become foolish in order to be wise.' Let this statement stand, for it is God's will in heaven that He has intended through foolishness to create wisdom; through wickedness to create the good; through sin to create righteousness; through folly, even through sickness to create health; through heresy to create churchliness; through unbelief the believer; and through the form of the devil to create godly people. You ask "How?" It shall be answered briefly and quickly. You cannot become before Go someone that you would like to be if you first have not become before yourself and before others the kind of person He wants you to be. God does intend, however, that you should become before yourself and others what you really are--namely, a sinner; bad, sickly, perverse, and devilish. Those are your names. Those are the things that you are in truth and they are your humiliation. As soon as that happens you are already before God what you wanted to be: holy, good, true, straight, and pious. On this basis you become a new person before yourself, others, and before God. Why are you surprised? Why are you bothered when you displease yourself and others? Because if you don't displease them, then you can't please God.
It helps our argument in this case that a serpent on a stick is later in the bible a sign of idolatry rather than trust in God. "Even heretical," Luther writes. It is God's command that Moses follows in putting the serpent on the stick. Look at it, and be healed.
Who wants to look at their sin in the way Luther presents it here? Isn't it bad enough the sin already bit us? Are we to look to it for our cure as well? And yet, Christ became sin for us, he is there on the cross lifted up and dying. We have been bit by sin, but we have only one recourse, to look to the very outcome of sin--death of the Son of God on the cross--as the healing for our sin.