1) It is written in the Zohar that the early pious men stayed awake all night on the night of Shavuos and delved into the Torah. Nowadays, most of the people who can learn, stay awake. It is quoted in the name of the Ar”I Zal that whoever does not sleep on this night and instead immerses himself in Torah, will live out the year, and no bad things will happen to him. (Mishne Berurah 494)

2) According to the simple explanations, the reson for the above custom is that during this night Klal Yisrael slept and Hashem had to wake them up to receive the Torah in the morning, and in order to repair this sin, we stay awake in the night and occupy ourselves with Torah. (ibid)