It means exactly what it says. He went to the his own people and they would not receive him. The Jews - his own people - rejected him.
The mention of “dogs” was a reflection of the common belief among Jews that non Jews were unclean, no better than animals. Jesus said this to test the Caananite woman who asked Jesus to exorcise a demon from her daughter. He also did this as an example to the Jews and present who accused his followers of not following the Jewish law, of eating with unclean hands. Jesus acted just like the Jews, stating that non Jews are dogs and that he was sent to help only the Jews. But then unlike the Jews, he turns around and grants the non-Jewish woman’s request because of her faith. It is a lesson to those around him that just because you are an ethnic Jew doesn’t mean that you are special, that faith in God is what matters.
Now among those who went up to worship at the feast were some Greeks. So these came to Philip, who was from Beth-sa’ida in Galilee, and said to him, “Sir, we wish to see Jesus.” Philip went and told Andrew; Andrew went with Philip and they told Jesus. And Jesus answered them, "The hour has come for the Son of man to be glorified. (John 12:20-23)
Jesus knows that his work is almost complete because Greeks - non-Jews - are starting to receive his message. The indication that his earthly work is almost done and that it is time to enter into his passion is when his teaching reaches non-Jews. As soon as non-Jews start asking to see him he knows that it is his time.