When it comes to current events involving birth control and the teachings of the Catholic Church you don't have to look far to find people saying things like "it's none of their business", "why doesn't the Church move in to the 21st century", "what does an old celibate guy know about sex anyway", "the Church just wants to tell women what they can do with their bodies" etc, etc ad nauseum. Sadly, it is not only people who are outside of the Church saying this but Catholics themselves often express like sentiments.
I think the most common objections to the Churches teaching are answered in Section 17 "Consequences of Artificial Methods" which states
"Not much experience is needed to be fully aware of human weakness and to understand that human beings—and especially the young, who are so exposed to temptation—need incentives to keep the moral law, and it is an evil thing to make it easy for them to break that law. Another effect that gives cause for alarm is that a man who grows accustomed to the use of contraceptive methods may forget the reverence due to a woman, and, disregarding her physical and emotional equilibrium, reduce her to being a mere instrument for the satisfaction of his own desires, no longer considering her as his partner whom he should surround with care and affection."There are two take always for me from this section. First, the real possibility of pregnancy through un-contracepted sex is an incentive to the young and unmarried to adhere to moral law. Namely that the sexual act is the right of a man and woman united in marriage for the procreation of the human race and the strengthening of the bond between them and that removing that incentive thereby making it easy for them to break that law is evil. Far to often our society tells us that providing kids with contraception is a good thing. They are going to have sex anyway, right? But no, as faithful Catholics we are not free to believe that. Pope Paul VI tells us that it is in fact evil.
The second take away was the Churches view of the woman. By standing firm in it's rejection of artificial birth control the Church is in fact affirming the real and true value of the woman. The Church is saying that men owe her reverence and care for her physical and emotional well being. By rejecting artificial birth control the Church is protecting the dignity of the woman, not controlling her.
I feel like the time I spent to read this document was well worth it. Humanae Vitae is only 12 pages long. I am a pretty slow reader and I like to go back over sentences and paragraphs several times while reading and I finished it in about 45 minutes. It has a lot of good teaching in it and would be a great read for someone who has wondered why the Church refuses to go along with the rest of society affirming the "goodness" of birth control.
The full encyclical can be found in pdf form here