A lot of people have already commented on why people turn to Buddhist tenents and they are absolutely correct. However, I would like to briefly explain some of the more concrete aspects of Buddhism that help followers try to reach enlightenment.
What you need to understand are some basics. Buddhism is refreshingly clear about how to cope with, and sometimes overcome, life's obstacles.
First, there are the Four Noble Truths:
1) life is suffering, (2) suffering is due to attachment, (3) attachment can be overcome, and (4) there is a path for accomplishing this.
The "path" is known as the "Eight Fold Path." Each of the "folds" means following a significant change in thinking. One changes from the "wrong" way, which is material/acquistion based thinking, to a "right" way of thinking in all manners of one's life. Those right ways are in thinking, intentions, speech, action, livelihood, effort, mindfulness, and concentration.
You should also know that the Buddha is not a god, but a man who has reached the most enlightened state ever known. Buddhists believe in reincarnation, so there are endless chances to get it "right."
A lot of Americans who adopt Buddhism view it as a philosophy rather than a religion. Many seem to like the idea of a religious system that doesn't require dogmatic belief in gods or heavens or a rigid set of rules. They view Buddhism as teaching how to free oneself from suffering and desire, how to overcome attachments and find peace, without necessarily needing to believe in some supernatural reality.
And the use of methods of meditation to introspect and develop one's awareness is a plus. Different religions use meditation to greater or lesser degrees, and one of the attractions of Buddhism is the use of meditation in a way that can be divorced from other kinds of religious belief.
I don't know think this is necessarily the case in cultures where Buddhism has been traditionally rooted. In many places, the practice of Buddhism has long been tied to belief in the culture's gods or spirits. There are times in history when Buddhism has been transplanted without some of its more supernatural or dogmatic beliefs, but that doesn't mean versions of Buddhism don't have those beliefs. I think a prominent one is the belief in karma and rebirth - cornerstones of traditional Buddhist belief.
I lived in South Korea for a year teaching English as a Second Language. There were churches on every corner, and Buddhist temples on every mountain. Having visited many temples, I must say that the beauty of the scenery, the serene nature of the monks, and the spirituality that was evident is a strong draw. However, as a Christian, I don't think I could find it comforting to believe in living multiple lives and moving up or down on the status scale depending on the way you lived the past life...will you be a tree or a rat? A cow or a stone? Will you know if you are a rat or a stone that you messed up your last life? If you know this, will you be able to live the next life in such a way as to climb higher on the rung or will you be so stressed about every move you make that you never progress? It seems very self-serving and not full of the hope and faith on which I have come to rely.
What intrigues me about the Buddhist religion is the sense of "self" and peace that its practitioners have found.
I was raised in the Christian tradition and followed it's tenants for most of my life, until I fell on hard times and was rejected by many of the parishioners I had worshipped side by side with for years.
Seeking an answer, I joined a support group where my mentor practices the Buddhist faith. As part of that, she has learned that we cannot control anyone around us but ourselves. She has learned to be grateful for the good things that are found in each day (even the bad ones), and she believes that we need to be good to and take care of ourselves...because if we are empty we cannot give to those around us.
As a Christian I was always taught to put myself last...which led to some pretty severe depression. When I learned to have a more "Que Sera Sera" attitude, my life became much more enjoyable.
In the 1960s and 70s many Americans explored alternative philosophies of life and afterlife, and I think we are entering a phase where that may happen again. I think some Americans like the idea that there is wisdom in Eastern religious thought and from perspectives that they hadn't considered before. I think the peaceful message of Buddhism, along with its emphasis on self-improvement and awareness is refreshing to some people.
There is a definite sense in which the simplicity of life that following Buddhism promises is something that I imagine most Americans would find appealing based on the consumer-driven lifestyle that so many have. In a society and culture where people have so much and there are so many choices, deliberately following a belief system that will give you a simpler life is something that many must find attractive.
My daughter in law is a practicing Buddhist. Also, as a veteran of the sixties in this country; I think that Buddhism's emphasis on renouncing desire; living a "right life" with respect to others is the most appealing element. Americans by and large tend to be a rather materialistic society. Those who embrace the teachings of Buddha tend to reject this materialism in search of greater contentment in life. It allows one contentment and ease of conscience with otherwise would be absent.
I would argue that the main thing that many Americans find attractive about Buddhism is its focus on the renunciation of desire. America today is such a materialistic society that this aspect of Buddhism would seem appealing to many people.
In the US today, people are typically involved in a "rat race" trying to gain more wealth and/or more status. This can be very psychologicall and emotionally wearing on the people who participate. Buddhism argues that people should renounce these kinds of desires. It says that people need to simply accept the world as it is, rather than always wanting things that they do not have. If your whole society is caught up in a "rat race" in which most people are striving for things they don't have, ideas like those of Buddhism could seem very attractive to many people.
Buddhism appeals to Americans, and to Westerners generally, because it offers a release from the dominant Christianity. If you look at the origin of Buddhism you'll find that its appeal was that it offered an escape from the dominant Hinduism. Wherever there has been a too rigid, or hierarchical system of belief, it's freedom-offering teachings such as Buddhism that capture the discontents. Within Islam, Sufism played a similar role. In the case of Buddhism, though, over the centuries it has itself developed orthodoxies, and has become just another organized religion. To Americans this is not obvious at first sight because there is the novelty factor involved. And when Buddhists seek to promote their religion they emphasize only the early pioneering and maverick spirit of the Buddha. Over a period of time the converts begin to see that one religion is no different fron another. Whether it's churches or temples, the institutions take precedence over individuality, which was the Buddhas key teaching.