The traditionally labelled second and third antitheses are closely related because they both deepen the Law such that certain behaviours are now considered adultery or precursors to adultery. The section begins:
“You have heard that it was said, ‘You shall not commit adultery.’ But I say to you that every one who looks at a woman lustfully has already committed adultery with her in his heart." (vv. 27-28)
Just as anger was in some sense equivalent to murder in the heart, so too lust is equivalent to adultery in the heart. There is a definite logic behind this idea: adultery is sex outside the marital bonds, and lust is desire for sex outside marital bonds. Or to express it another way, adultery is an inordinate act of sex, and lust is the inordinate desire for sex.
Adultery, as I have said before, is necessarily a breaking of the marital bonds with sex outside of them, and lust-as-adultery-of-the-heart is therefore also in the context of marriage. Nonetheless, lust outside marriage is also sinful, but it becomes lust-as-fornication-in-the-heart, not adultery of the heart. That means that lust even after one's future wife is sinful.
How does one cope with something that seems as innate as a desire? Jesus continues:
your right eye causes you to sin, pluck it out and throw it away; it is
better that you lose one of your members than that your whole body be
thrown into hell. And
if your right hand causes you to sin, cut it off and throw it away; it
is better that you lose one of your members than that your whole body go
into hell." (vv. 29-30)
I think this is clearly hyperbole to make a point, but it is a forceful point nonetheless. Jesus' advice appeals to priorities: the eternal matters more than the temporal. It is worth gorging out an eye if it causes you to sin, because sin kills and condemns eternally, whereas it is only temporarily that one would be without an eye. If the problem were really solved by taking out the eye, then it would be worth it.
Now, a quick word on hell. It is interesting that people complain about God in the Old Testament scriptures being a good of anger and judgement, and Jesus the revelation of God as loving, kind and nice, because this is far from the case. One has to cherry-pick from the Old Testament to find God as solely angry, but one also has to cherry-pick from the New Testament to find God as solely nice. The fact is, Jesus speaks about hell more than the whole Old Testament, as far as I can tell. This is not surprising, since Jesus as "the Lamb of God who takes away the sins of the world," is keenly aware of the consequences involved with sin. Nonetheless, the use of the imagery of fire is not exactly literal, not exactly metaphorical - fires of hell refers to, in the Greek, the Valley of Gehenna, a place south of Jerusalem which served as a large dumb, where rubbish was burned continuously. It is a real place that Jesus refers to, but what we call hell is not literally that place south of Jerusalem, particularly because such a place no longer exists. Perhaps we should take this as indication that hell no longer exists! More seriously, this talk of hell is borrowing imagery to make a point, not speaking of a real pit of fire.
How does one apply this? One is meant to see how intensely important it is to avoid hell. If going to the bar with one's friends might cause drunkenness, then it is better to cut off those friendships (gathering in that particular context) than go to hell. If walking down certain streets makes on burn with lust, but it is the only way to get to work on time and avoid being fired, then it is better to lose one's job than one's soul.
Is that not too radical? Well, it depends on whether one thinks damnation is a big problem or something of relatively small dimension. Jesus' argument rests on this point: that no cost is too great for avoiding hell.
This little section is actually quite amusing, Jesus is not actually illustrating how to deal with lust and adultery as much as he is setting up the principle that things must be removed if they cause sin. The implication seems to be that, if one is tempted to commit adultery, one should cut off the parts involved in adultery. This is a crowd which is made up of mostly men, I suspect, so the idea is "cut off even your testes and penis before you would commit adultery." See how Jesus can also be funny, in his own dark way?
“It was also said, ‘Whoever divorces his wife, let him give her a certificate of divorce.’ But I say to you that every one who divorces his wife, except on the ground of unchastity, makes her an adulteress; and whoever marries a divorced woman commits adultery." (vv. 31-32)
Here, I remind that the Law was a national law, which means that at times the morality was made more lax for the sake of the people's ability to function. Now Jesus is revealing, or making explicit, the fullness of the moral law, where divorce is not permitted (except on the grounds of unchastity). There is actually some debate about whether the Greek is correctly understood as "unchastity," but I will have to trust the translators since I am not able to judge for myself.
From this section alone, Jesus explains the moral law as saying that no longer may it be deemed fine to divorce except under very particular circumstances. The companion passage on divorce in St Matthew's writings is Matthew 19, but let me not confuse teachings, and centre primarily on this text here.
Jesus' rationale is interesting, because it is not immediately obvious why divorce makes the wife an adulteress. Is it because she will be forced to re-marry, since at that time being a single woman was practically impossible? I think this is the most likely explanation. Why is it unchastity? Very simple: divorce does not really exist. Issuing a certificate of divorce cannot separate the man and woman united in marriage, because no such thing can annul a covenant. So when one attempts to remarry and begins marital relations, what happens? One is having sex outside true wedlock, and is therefore committing adultery. That is why whoever marries a divorced woman commits adultery - she is the wife of another person.
The interesting consequence of this is that marrying a woman who was divorced, but whose husband has now died, is no longer adultery under that logic. The ending of the covenant of marriage at death was the idea that St Paul appeals to in Romans 7 - although with Jesus conquering death, it is not entirely obvious that now marriage can endure and conquer death too.
How should this play out in the Church and in the world? Well, it is my opinion that the defenders of marriage as it has universally been understood (between members of opposite sex), throughout most cultures and times, lost their battle when they allowed divorce. What demeans the institution more, hitherto unheard of unions or complete freedom to break off at whim? It is my contention that secular legalization of divorce is a far greater evil, has and will continue to produce far more broken families, children without both parents, than same-sex marriage will. Does that mean same-sex marriage is just fine, since marriage is already broken? It does not follow in the slightest. Still, one's platform is backwards if one is pro-divorce but against same-sex marriage.
Amidst all the wordliness that has crept into Christendom, it is reassuring that the Church has stuck by her commitment to the truth, even though all of England went Anglican because she would not compromise and let a randy monarch have a divorce, and even though many hate her for it.
 If this is the correct analogy being made, then it follows that desire for sexuality after one's own husband or wife is not lust, and not sinful.
 Jesus will later say that divorce was permitted because of the hardness of heart of the Israelites - permitted in Deuteronomy, though, which is the second law.