The holiday of Shavuot commemorates when the Israelites received the laws of the Torah on Sinai. Exodus 19:10, immediately before the Ten Commandments are revealed, has God say to Moses that they must be pure in order for them to be kept.
After this same quote, God gives reason and judgement as to why we must obey him. God wants all the people to be clean and pure in order for them to come into their own; in order for you to take something to heart, you have to be rid of all doubts of truth or not. God doesn't want them to touch the mountain because he knows you have to work and give space in order for benefits to come to you. As the adage says, "Good things come to those who wait". The Israelites understand this and thus wash their clothes so they can be pure and accepting of his command when he is ready.
God is ready for one of the biggest moments in Exodus, the giving of the Ten Commandments in Exodus 19:18. The significance of Moses' words is profound in Exodus 19:23, people respecting God. The primeval history is all about disrespect of God, from the Fall of Man to Noah's Ark. It is only after Abraham founds the Jewish people that respect for God first happens (the Binding of Issac). Moses culminates Abraham's foundation by respecting the words of God, which is touched upon greatly in Leviticus.
The final two verses of this chapter affirm that certain people (those who have always been loyal to Adonai) are detailed to certain privileges that other wishy-washy people aren't. God thinks that if the Israelites see him, he will be treated like an idol (see 10 Commandments and Ki Tisa) and that's why only Moses and Aaron are allowed to see him in the giving of the Ten Commandments.
The later portions of Exodus 19 result because of Bo, Beshelach, and the exodus from Egypt. One of the most important phrases of this chapter is the 8th verse where the Israelites are thankful for God guiding them to the Holy Land. And because of this, God forgives mistakes, tries to do right to the Jews, and allows the purpose of Shavuot to be in existence.
CONCLUSION: Exodus 19 is a very important chapter in the Tanakh. It holds many key words of wisdom, foreshadows later events, is an amazing prelude to Exodus 20 (and the revelation at Sinai), and is just really GOOD!