Back in Jesus’ days, no one used the expression “children of God” about anyone else (the jewish people were known collectively as “the people of God”). That’s why they were so offended when Jesus claimed to be the son of God.

They understood exactly what he meant. If he was saying he was the son of God, he meant that he was born of God, that he was the same as God.

Take a look at these verses (emphasis added by me):

But after he had considered this, an angel of the Lord appeared to him in a dream and said, “Joseph son of David, do not be afraid to take Mary home as your wife, because what is conceived in her is from the Holy Spirit. She will give birth to a son, and you are to give him the name Jesus, because he will save his people from their sins.”
All this took place to fulfill what the Lord had said through the prophet: “The virgin will conceive and give birth to a son, and they will call him Immanuel” (which means “God with us”). (Matthew 1:20-23)

“I and the Father are one.” (John 10:30)

“Why then do you accuse me of blasphemy because I said, ‘I am God’s Son’?” (John 10:36)

Again the high priest asked him, “Are you the Messiah, the Son of the Blessed One?”
“I am,” said Jesus. “And you will see the Son of Man sitting at the right hand of the Mighty One and coming on the clouds of heaven.” (Mark 14:61,62)

These are just a few quotes, you will find many more in the Gospels and all throughout the New Testament. For example, the well known verse of John 3:16 says (emphasis added by me):

For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.

People understood what Jesus meant, that he claimed to be God, and they accused him of blasphemy:

“We are not stoning you for any good work,” they replied, “but for blasphemy, because you, a mere man, claim to be God.” (John 10:33)

If Jesus was the only child of God, then how did common people got to be called “children of God”?

Yet to all who did receive him, to those who believed in his name, he gave the right to become children of God— children born not of natural descent, nor of human decision or a husband’s will, but born of God. (John 1:12,13)

So in Christ Jesus you are all children of God through faith, for all of you who were baptized into Christ have clothed yourselves with Christ. (Galatians 3:26,27)

The answer is: through Jesus himself, by receiving him, having faith in him and believing in his name. (I will write more on this subject later.)

Jesus was conceived by the Holy Spirit, he is the legitimate son of God. And through faith in him we are given a new spiritual birth (represented by the baptism) that turns us into “children of God”. That allows us to legitimately call God “Our Father”.