If Matthew emphasizes Jesus as king, Luke highlights the humility of Jesus' birth. Rather than being welcomed by kings, Jesus is greeted by shepherds, some of the lowest in society. Luke focuses on the role of women in the story, individuals with low status in that world. One of these is the prophetess Anna, who had been a widow perhaps for as much as sixty years--again, individuals usually overlooked and disempowered in that world.
We are so used to splicing together these "Christmas stories" that we often miss that each of the gospels has something unique to contribute to our understanding of Jesus. It's interesting to wonder how our Christmas plays would be different if we only had Matthew or only had Luke. For example, if we only had Matthew, we would think Jesus was originally from Bethlehem and only later moved to Nazareth in the north to escape persecution (Matt. 2:1, 22-23).
Luke begins with the birth of John the Baptist. We would not know from Matthew, Mark, or John that Jesus and John the Baptist had any family connection. From Luke we also learn that John the Baptist is of priestly lineage, something we are not told in any other gospel. It is intriguing to speculate if John was an Essene or if some early believers had been Essenes. John baptized not far from the most famous Essene location on the northwest side of the Dead Sea.
The Essenes were the third of the three best known religious groups among the Jews of Israel, the others being the Pharisees and the Sadducees.  Until the discovery of the Dead Sea Scrolls, we knew of them primarily from some scattered comments in ancient writings like those of the Jewish historian Josephus and the Jewish thinker Philo. Since the Dead Sea Scrolls were probably the library of an Essene community, we have since been able to form a more extensive sense of who they were.
There seem to have been different groups of Essenes, some of whom were stricter than others (just as the Pharisees had different schools as well). Some married and lived in cities. Others separated themselves from society and were celibate. At least some of them were much stricter than your average Pharisee. For example, the community at Qumran by the Dead Sea viewed itself as a substitute for the temple while the temple was impure and they lived by the purity standard of priests.
Their sense of purity seems to be closer to the Sadducees than to the Pharisees, which makes sense if they had a significant priestly background at some point of their history. They seemed to have shared their possessions in common. They probably had a more "apocalyptic" outlook on soon coming events in which God would judge the world and those in the true assembly or church would be saved. They perhaps had a stronger sense of eternal punishment and hell than other Jewish groups. They have also provided us with some of the clearest examples of the expectation that a Messiah was coming to restore and purify Israel.
Another feature of Luke's birth story are the number of songs in them...
 The majority position of experts is that Matthew and Luke are independent of each other but used common sources like Mark and another collection of Jesus' sayings. However, it would explain a few other similarities if, in addition to these sources, Luke also knew about Matthew's gospel. See Mark Goodacre, Questioning Q, ***. The real answer to how it happened may be more complicated than any of us would imagine.
 One of the best overviews of Judaism in the lead up to Jesus is James VanderKam's An Introduction to Early Judaism ***.