What is the meaning of the bronze serpent used to save the Israelites?

The bronze serpent represents the provision made by God. They, by themselves, would not be able to recover from the deadly wound, but through faith only in what God has provided as the solution can they be healed. It represents Christ, the grace that comes only through Christ.

John 3:14-15 (NKJV) And as Moses lifted up the serpent in the wilderness, even so must the Son of Man be lifted up, that whoever believes in Him should not perish but have eternal life.

The book "Patriarchs and Prophets" by Ellen White offers a well-written explanation:

The lifting up of the brazen serpent was to teach Israel an important lesson. They could not save themselves from the fatal effect of the poison in their wounds. God alone was able to heal them. Yet they were required to show their faith in the provision which He had made. They must look in order to live. It was their faith that was acceptable with God, and by looking upon the serpent their faith was shown.

"As Moses lifted up the serpent in the wilderness," even so was the Son of man "lifted up: that whosoever believeth in Him should not perish, but have eternal life." John 3:14, 15. All who have ever lived upon the earth have felt the deadly sting of "that old serpent, called the devil, and Satan." Revelation 12:9. The fatal effects of sin can be removed only by the provision that God has made. The Israelites saved their lives by looking upon the uplifted serpent. That look implied faith. They lived because they believed God's word, and trusted in the means provided for their recovery. So the sinner may look to Christ, and live. He receives pardon through faith in the atoning sacrifice. Unlike the inert and lifeless symbol, Christ has power and virtue in Himself to heal the repenting sinner.

http://www.whiteestate.org/books/pp/pp38.html

Jesus refered to what He was going to do, as a fulfillment of the true meaning of the bronze serpent being lifted by Moses in the wilderness (John 3:14-15).

¹⁴And as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up: ¹⁵ That whosoever believeth in him should not perish, but have eternal life. (John 3:14-15)

Similarities:

• The Israelites were bitten by venomous snakes and were going to die // all people are 'bitten' by the Serpent and due to sin are going to die (the wages of sin is death)

• God provided a way that people could not die due to the snake bites // God provided a way that people can be saved and not die due to their sin

• The snakes were the cause of the problem (sin) and the bronze snake was the solution // Jesus was sinless, yet on the cross He was made sin for those that would believe on Him (see 2 Cor. 5:21), baring in Himself their sins

• The lifted bronze snake was foolishness for some of the Isrealites, because they couldn't see what good is a statue of a snake when they had actual snake bites (and those died) but those that looked up to it with faith, lived // The cross of Christ is foolishness to those that perish, but to those who look up to it with faith, it is the power of God unto salvation! (see 1 Cor. 1:18)

Due to all those and other similarities, Jesus used the event with the serpent as an illustration, so that Nicodemus (a teacher of Israel, well aware of this event) could understand who Jesus truly was.

In brief the bronze serpent is an Old Testament image of the cross of Christ, and what He bore on it!

However we must be careful to worship Christ and not the cross, because many years after the event in the wilderness, some Isrealites started worshipping the bronze serpent (as an idol), as if it had saved them, when in fact it was God who saved them, using that means. (see 2 Kings 18:4)