First and foremost, serving as a foundation for all the other issues here, is the issue of why one should worship God at all. Many people have given suggestions over the years, and it's hard to say which might be convincing to any individual. A common theory is that as creations of God, we should worship the creator. I haven't read any philosophical works of Christian existentialism, but since this line of reasoning is reminding me in many ways of Sartre, I wonder if this is what is meant? When an individual creates a thing, the thing has as its very reason for being service to its creator. There is an assumption that since God created us, He has a right to dictate our purpose in life. The idea that that purpose is worship of God is a secondary assumption that is built on a more complicated religious framework; a deist might agree that the Creator has a purpose in mind for us, but not necessarily agree that the purpose is known. As a person who believes in the Bible, though, one can read the Bible to get glimpses of what the purpose might be.
While many theists believe the fact that God is our creator is enough reason for Him to deserve worship, they will be quick to point out that that is not the only reason, nor even the primary one. While God will from time to time point out the fact that He is the creator of everything that exists, He is far more interested in reminding people of things he has done since creation. In the Ten Commandments, He doesn't mention creation, but instead points out that He is "...the LORD thy God, which have brought thee out of the land of Egypt, out of the house of bondage." God's primary reason that He gives for worshiping Him is His acts of salvation. He saved the Israelites from slavery in Egypt, He brought them out of their exile in Babylon/Persia, and in the end, He sends a Messiah to save them and the whole world from their sins. Why give allegiance and worship to God? Because He's constantly looking out for the well-being of the people He created. As Jesus said in John 10:10,
"The thief cometh not, but for to steal, and to kill, and to destroy: I am come that they might have life, and that they might have it more abundantly."The "thief" is, in this context, anyone who claims to have a person's best interests in mind, but really means to take away every good thing that God has given them or promised them in the future. (And no, this does not mean the O.T. prophets, as is implied in the note on verse 8.) It's a very common theme throughout the Bible that following anyone other than the one true God will lead to a loss of prosperity, happiness, life and salvation.
So, once one accepts the idea that following the God of Israel is a good thing, then come the guidelines.
Avoiding idolatry? This is not actually a specific matter of worshiping a false God; there is an idea involved here that as pagan nations surrounding Israel filled their temples with statues that represented their gods, the Israelites might be tempted to do the same thing. Actually, they did. In Exodus 32, there is the famous story of the golden calf. In the story, note that Aaron does not give the calf a funny name and make up a god, they're all taking this statue as being a representation of the deity that brought them out of Egypt! The problem with idolatry is that once God is represented by a hunk of metal, He gets mentally reduced to a hunk of metal, and who has respect for a hunk of metal? This is actually a common theme in many religions, including Hinduism* in some ways, surprisingly: the idea that the deity is so vast and powerful that He/She/It cannot be reduced to a mere representation, but must be left abstract. The Israelites quickly lose their moral focus once God is brought down to a finite representation before them. It should be noted that despite the SAB note in the beginning of that chapter, the only thing Aaron told them to do was give up their earrings. (Not that Aaron is at all faultless here; Moses and God put a lot of the responsibility on him. I'm sure I'll need to address that chapter in specific later on.)
What of apostasy? God puts out a lot of warnings against the Israelites ever turning to other Gods, setting penalties on people who do so or encourage others to do so. Later on, when they take the Promised Land, they drive out all the occupants for numerous reasons, the main one probably being that they are pagans. Well, that's more of a straightforward matter of what I was talking about above; while the true God gives "life, and...more abundantly," other so-called "gods" have either nothing to offer or they have destruction. I'm not completely certain, but it may certainly be the case that God is more likely to punish apostates than infidels. It's one thing to simply not know that the God of Israel is the one true God, and another to know this and willingly turn away. I think that this may be what Paul is talking about in Romans 2, and Jesus hints at a few times. If you know the right thing to do with regards to God, and refuse to do it, how can you stand in comparison to those who were essentially ignorant of the truth, but did the right thing anyway?
The topic of blasphemy is a tough one, because it has many different layers to it, perhaps more so in Judaism than in most other religions. As demonstrated with a humorous effect in the movie Life of Brian, even saying God's name out loud was considered blasphemy, and to this day, observant Jews will not say "the Name" as they refer to it. Why? Because God is considered so great, so majestic and so holy that it simply is not possible to utter His Name with enough reverence to measure up. Now the more modern Christian notion of taking the Lord's name in vain meaning referring to God in a flippant matter, or worse, using "God" or "Jesus" as a curse word, seems downright tame in comparison to the Jewish approach. But indeed if God is so very Holy that even angels are hesitant to look at Him, His Name ought to be treated with a great deal of respect, far more than even those who consider themselves His followers give it. I talked a bit about how especially in the Bible, a person's "name" was synonymous with their reputation, and who could have a higher reputation than the Creator of the universe and Saviour of humanity?
Anyway, as I have suggested in other places on a handful of other issues, it may be in a large part the case that by God's very nature and His Holiness that what in our original verse sounds like a threat is merely a statement of fact. "Have respect for the law of God or face dire consequences!" may be in a theological sense equivalent to "Have respect for the law of gravity or face dire consequences!" It simply naturally follows.
(*Contrary to outside appearances, Hinduism is, at its root, a monotheistic religion. All of the different deities worshiped in South Asia are in some sense considered to be in actuality manifestations of a singular higher power known as "Brahman".)