How can Milton's Areopagitica be summarized?

Areopagitica is arguably one of the most important defenses of free speech in the English language. The title comes from the Greek word Areopagitikos which refers to a speech given by the fifth-century BC orator Isocrates on the hill of Areopagus in Athens.
The occasion of Areopagitica was the passing by the Puritan Parliament of the Licensing Order of 1643. This brought publishing under the official control of the government. Anyone publishing written work would have to submit it to government censors who would then decide whether or not it was fit to be published.
Milton had been a staunch supporter of the Parliamentary cause during the Civil War, but the issue of censorship prompted a growing disillusion with the direction that the Presbyterians in Parliament seemed to be taking the country. For Milton, as both a writer and a devoted Protestant, censorship was something one associated with the crypto-Catholic regime of Charles I or the officially Catholic governments of Europe. To him, it had no place in a godly Protestant republic.
Milton criticizes censorship and defends free speech on a number of grounds: some practical, some philosophical, others theological. He passionately asserts that God gave human beings reason, and with that reason came the freedom to choose. If you take away that freedom, as censorship does, then not only are you going against God's will, but you don't even remove the desire for freedom, which lingers on regardless:

"Though ye take from a covetous man all his treasure, he has yet one jewel left: ye cannot bereave him of his covetousness."

Even if you take a man's riches from him, he'll still desire them; it's the same with opinions. Censoring them won't make them go away.
Freedom of opinion is important because without it there can be no increase in knowledge or advancement in learning:

"Where there is much desire to learn, there of necessity will be much arguing, much writing, many opinions; for opinion in good men is but knowledge in the making."

Opinions should be voiced freely, even plainly incorrect ones. Good and evil are closely intertwined, as Milton so wonderfully showed in Paradise Lost. It was only through eating the fruit of the Tree of Knowledge that Adam and Eve eventually understood the difference between good and evil.
Furthermore, Milton argues that for a book to be officially sanctioned is to slap a badge of dishonor upon it:

"[W]hen every acute reader, upon the first sight of a pedantic license, will be ready with these like words to ding the book a quoit's distance from him: "I hate a pupil teacher; I endure not an instructor that comes to me under the wardship of an overseeing fist. I know nothing of the licenser, but that I have his own hand here for his arrogance; who shall warrant me his judgment?"

As soon as we see that a book has been officially licensed by the government, then as discerning readers we'll want nothing to do with it. The state may have the right to govern, but it has no right to censor thoughts. And by censoring what people can and can't say, it actually undermines the state's proper authority by bringing it into disrepute.
Milton believes that the truth will always ultimately come out. But it can only do so if false opinions are given as much free reign as true ones. Truth doesn't just suddenly drop down from heaven fully-formed; it needs to be fought for, and that can only happen if there is freedom of opinion:

And though all the winds of doctrine were let loose to play upon the earth, so Truth be in the field, we do injuriously by licensing and prohibiting to misdoubt her strength. Let her and Falsehood grapple; who ever knew Truth put to the worse in a free and open encounter? Her confuting is the best and surest suppressing.

In a contest between truth and falsehood there can only be one winner. So why should we even think of censoring false opinions? The best way to suppress falsehood is not through state-sponsored censorship but by exposing it to the light of truth. Milton rightly senses that the Presbyterians in Parliament who support censorship are somewhat insecure in their opinions and so see censorship as the only way to defend them.
Finally, Milton argues that if you start censoring books and opinions then there's no telling where it might lead:

If we think to regulate printing, thereby to rectify manners, we must regulate all recreations and pastimes, all that is delightful to man. No music must be heard, no song be set or sung, but what is grave and Doric. There must be licensing of dancers, that no gesture, motion, or deportment be taught our youth, but what by their allowance shall be thought honest . . . 

Parliament has embarked upon a dangerous path. The crude extirpation of falsehood by the blunt instrument of censorship can just as easily be applied to other forms of human expression. Where will it all end?