That, really, they might should have refused. After all, much of the remaining Old Testament narrative relates how the Israelites were frequently unwilling to fulfill their side of the covenant that they enter in Exodus 19 (link). However, that's not what happens here.
Instead, what happens is this: the Israelites come to Mount Sinai. Textually speaking, they come to the wilderness of Sinai and camp in front of the mountain, and that mountain we call Mount Sinai.
The Lord God speaks to Moses and states that if the people of Israel will obey His voice and keep His covenant, then they will be His possession among all the peoples. He will be their God, nearer to them than to all other people. It is, though, a conditional covenant.
There are two basic forms of covenants in life. Since those two forms reflect in Scripture, let's look at them both.
The first is the unilateral covenant. This is a one-sided commitment, whereby one party states that they will do something or perform a specific action with no regard for the behavior of the other party. This is typically a gift: party A will do X for party B. Does not matter what B does.
The second is the bilateral or conditional covenant. This is the more common two-sided commitment, whereby one party states that if the other does their actions, they will do other specific actions. This is a business covenant: party A will do X in response to party B action of Y.
Now, that's a highly simplified view, but it's adequate for what we see in Exodus 19. We see the bilateral covenant. God is promising that if Israel (party B) obeys the Word (action Y) then God (party A) will be His special possession (action X). The trick with a bilateral covenant is that it puts the responsibility on party B. They must do action Y to receive the benefits of the covenant, or else A is off the hook.
So, God offers this covenant to Israel. He does not offer it as merely a generic "god," though. This is offered by God using His specific, covenant name (see discussion here, especially at the end in the nerd note). That is an important point. This is not a covenant offered to serve whatever god you can come up with or feel like serving. This is a covenant with YHWH, the Most Holy, who Moses spoke with some years prior, who appeared to Abraham, Isaac, and Jacob, and who created all things.
This covenant, though, is not about just Israel. Take a look at Exodus 19:5-6 and see what they say. God declares His sovereignty over all the earth in this passage. The Israelites are to function, as they have for Midian and Egypt to this point, as God's priests who will spread His message to all people. They will be His possession so that the world, when seeking God, will know where to turn.
In short, the whole country is intended to be the shining light of the world through the Word of God. This is the covenant God puts before the people. It is not a covenant for the sake only of the Israelite people. It is a covenant for the sake of all mankind.
The process, though, is challenging to see through to its completion. We see God revealed here as dwelling amidst inapproachable fire, that His Holiness prevents any from coming near Him without His approval. Not even sacrifices are enough without His gracious calling of the people near to Him. That's important: many of the cultures around the Israelites from Egypt through the Promised Land had gods that dwelt in temples that people could enter—their God, though, was not to be found in one place that could be opened or closed as people desired.
Not a nerd note: When you think of the two types of covenants, what do you think a marriage covenant is? If you think it's bilateral, you're wrong. A marriage covenant is a unilateral covenant made between you and God regarding how you will treat your spouse. It is not a bilateral covenant between you and her, that you can drop if she ignores part of it. It is a covenant to God of how you will treat her. Ladies, the reverse is also true. You covenanted with God how you would treat your husband. Scripture gives points at which God may release you from that covenant, but it's not a trade-off with your wife or husband. You are bound to the exclusive nature of that covenant until God releases you from it according to His Word.
Today's Nerd Note: Point A: Exodus 19:23 looks like Moses has to remind God of something. This is not the case, and never is. God does not forget. Rather, Moses is restating for clarity and committing to obey.
Point B: Exodus 19:15 is a little phrase that slips into your Bible to make for uncomfortable moments in either family devotions or Jr. High Sunday School. It is an accurate reflection, though, of the Law that God will reveal. It does, also, deal with sexual activity. The Law provided that sexual activity led to ceremonial uncleanness. This was not sinful, rather just ceremonial. One did not indulge in the marriage bed and then go straight off to offer a sacrifice. Part of this would have been hygienic and part is focus.
If you've been sexually active, your mind has been on that rather than on anything else. (This should be the case: husbands, wives, your focus in that time is each other.) As such, it would be hard to come from that intimacy and shift gears rapidly to worship and sacrifice, especially to considering the gravity of sin after the moments of ecstasy.
So, it was not uncommon (1 Samuel 21) for men to abstain from sexual activity when involved in certain activities. Neither does this disappear in the New Testament: Paul speaks to the need for limited times of marital abstinence in 1 Corinthians 7.
The idea here is not that sexual activity within marriage is sinful. It is rather that this activity can be a distraction. Especially since the people at the time were to be waiting and ready for the call of God and the sound of the trumpet—it was no time to have another commitment to fulfill.