John 1:9-13 says that while the world did not recognize Him and His own people rejected Him, there are some who did receive and believe in Him. Doesn't this sound like a contradiction?

Certainly, the world (presumably Gentiles are in view) would not recognize Him, for they were godless. But the Jews, the people of God, who should have known better, even they rejected Him.

Then who are those that received and believed in Him? These are the ones Jesus says in John 1:12 who have been given the right to become children of God.

The answer is in John 1:13. The new birth is not possible by human effort. Those that receive and believe in Jesus did not do so by their own human effort but by God. That is why John says that the people who believed in Jesus were not born of natural descent, nor of human decision or a husband's will but born of God. God did it. So then, even though by nature everyone rejected Jesus, God intervened to save some. How? Look at the dialogue between Jesus and Nicodemus in John 3. There we discover that God did it through a rebirth, even a new birth, a birth from above, by the Holy Spirit. Being born again, people could both receive and believe in Jesus Christ, whereby they are then given the right to become children of God. This is the argument of sovereign election. People could not believe because they would not believe by their own free will; therefore, God had to supernaturally overcome their depraved free will by the regenerating work of the Holy Spirit, so that those, who by nature would turn away from God (cf. Ephesians 2:1-4; Romans 3:1-9), would be reborn with a new nature that both could and would turn to God in faith.