Astrology is a type of spirituality. It's not an organized religion, it's not a cult, and it's not science (although it has scientific elements - astronomy and math).
Astrology isn't about zodiac signs specifically. It's about the relationship of us as individuals, a collective group, and our place in the universe. I believe that our private universe (Sun, Merc, Ven, Earth, etc) is like it's very own bubble within the larger galaxy. Because of our "bubble," our sun, asteroids, planets, and life on Earth share the same rhythm.
On a smaller scale, you can picture this: Humans typically rise with the sun and go to sleep when the sun is down. Our bodies have evolved to do this for whatever reason. So our day to day rhythm is in harmony with the sun. And our bodies have evolved with the seasons as well. During winter, when it's usually rainy, cold, and overcast (because of the Earth's angle to the Sun at certain times of the year), people tend to become depressed. The Earth is also in harmony with the Moon because it controls our tides (which controls our weather, which has an effect on our moods). And it's been scientifically proven that a women's menstrual cycle is in sync with the cycle of the Moon.
Where the belief comes in with astrology, is when you cross over into believing that all the planets' cycles are affecting our body to such a degree that it shapes our behavior. The zodiac is just symbols. The constellation Cancer isn't literally affecting you. Cancer represents the Moon which is what's really affecting you.
Because astrology is about the relationship of the planets and not just simple zodiac signs, you can't really look at Sagittarius/Taurus relationship and say it's incompatible. You have to look at the relationship of their planets altogether. Maybe their Sun signs are incompatible, but maybe Her Moon is compatible with his Sun. Or her Venus is compatible with his Mars.
If you would like to get started on learning more about it, type in your birth information at astro.com. This will give you your Natal Chart. Then look up these key words:
1. Look up the Greek symbols for the planets. This will help you to read your Natal Chart.
2. Synastry Chart and Composite Chart
3. Ascendant/Rising Sign
4. All the planets' signs
6. Natal Chart houses and their meaning
7. All the zodiac signs and their meaning
8. All the planets' meanings
To type in a meaning of an aspect, you can type in something like: Mercury conjunct Uranus or Sagittarius in the 5th house. Then you will find articles describing what that possibly means for you. But you must always remember that it's about the relationship of all the planets and houses together. So if Mercury conjunct Uranus in the 11th house doesn't sound like you, then maybe you should look at the other planets and houses influencing those two planets. Astrology is more of a puzzle/code that you have to crack. The more you know about the world and the more you mature spiritually, the more you will know about astrology and vice versa.
There is no scientific proof of this. But for me personally, I've found it to be fairly accurate in depicting my personality and how I feel at times. You will know it works for you when you read it. If you find meaning in it, then it's true. If you don't, then it's not. Perception is reality and Astrology is a spiritual journey, not a paragraph in the back of a magazine.