If it looks like a duck, walks like a duck, and quacks like a duck - it's a duck
That's the easiest example of the Occam's razor, in action. Though this may sound trivial, consider this example.
Suppose we create an artificial robot which looks exactly like a duck. Upon observation of the above kind (watching it from a distance), Occam's razor would force us to conclude that what we are seeing is a duck (and not the duck robot). Surely, upon further investigation, we will know that it's not a duck (when we dissect it, we will see motors and circuits and not blood and flesh). Till we have that extra bit of data about the interior of the duck-robot, the simplest conclusion is that its a duck and that's Occam's razor.
Philosophers have spent ages on the following example, but let me try my bit as well. I apologize for putting on the pompous hat. "Why should we be ethical?" If that's posed as a question, one may answer: "We should obey god", and "God wants us to be ethical". This theory of ethics involves at least 3 axiomatic statements.
1. God exists
2. We should obey him
3. One of god's orders is to be ethical
Out of this, the first statement is not exactly axiomatic because god has to be defined when we say it exists. To my understanding 2 and 3 are pretty straight forward objective statements. On the other hand, if our answer is this in a rather complicated manner as Kant did : "Always act according to that maxim whose universality as a law you can at the same time will". For example, I should not bribe, because I do not want to live in a world where bribery is abundant. This justification of being ethical requires only one axiom or assumption compared to 3 for the god theory. Occam's razor would suggest that we do should not chose the god theory to justify ethics over the Kant theory. People now know better alternatives to the Kant theory as well. Interestingly, if god were an empirical fact of life like ducks and buffaloes, we will not need to hypothesize about it and then the god theory would be better. As I think about this, the chinese had developed a school of thought called the loyalist which was essentially:
1. The king exists
2. We should obey him
3. One of King's orders is to be ethical
Here, the king exists empirically in people's lives. Empirical existence loosely means tangible existence (lokayat in sanskrit). Hence on the face of it, the loyalist theory for ethics is equivalent to the Kantian theory.
The most beautiful example I know, of the success of the razor is due to Einstein. I am going to try and write it in the simplest language possible to me.
In 1864, the young James Clark Maxwell was writing equations to describe how magnets and charged balls attract each other. Just out of curiosity, he applied the same equations to space where there are no magnets and charged balls. Surprisingly, he found out that even in absence of magnets and charged balls, the fields produced by them can exist in a periodic fashion. This field travels in a single direction. From the mathematics of it, he could simply calculate the velocity of the field which turned out to be the measured velocity of light! Indeed, light was an electromagnetic wave. From magnets and pith balls mathematics lead Maxwell to the true nature of light! But there was one problem. As we know in common sense, waves require a medium, sound waves travel through air and water, vibrations take place on strings, sea waves are on water and so on. Maxwell's equations didn't require the light to travel through any particular medium, from the face of it, it seemed that they could exist even in vacuum! Blasphemy, though Maxwell and readily proposed a medium called 'ether' which pervades everything, just like the Brahma of Sankara. Bodies could move in the ether without any friction he said. This ether was required just to support our common sense that waves need 'something' to propagate.
We surely know that while traveling in a car, the fellow drivers appear to us to be slower than their actual speed, since they are traveling with us. But, when we slow down, they appear to be much faster! In short, the speed of the 'observed' depends on the speed of the 'observer'. Now, if our earth is moving through this ether, one can calculate the velocity of light in different ways so as to detect the motion of earth. After the famous Michaelson-Morley experiment to detect earth's motion in ether, it was found out that unlike the speeding car, speed of light is constant, independent of the 'observer'. This was a shocking observation, which defied all human intuition. To explain this rather challenging experiment, Lorentz came up with a theory more complicated than the Maxwell's equations, now known as Lorentz theory of ether. This theory made some unintuitive postulates and confirmed that the drift of earth in ether cannot be detected by measuring the speed of light. The only purpose of the mathematical jugglery was to come up with a framework which supports the theory for ether. In short but very importantly, Lorentz theory could explain all the available experimental observations. To follow the Occam's razor framework, Lorentz had proposed a theory which explained everything that was known about light at that time with some postulates regarding ether. This theory had axioms about the nature of ether and its interactions with matter and light.
Tired by this, Einstein in 1905 simply came up with an alternative (now better known as the special theory) where he didn't require at all, the mumbo-jumbo about ether. Instead, he made fundamental assumptions about the nature of light, viz. the principle of relativity and the principle of invariant light speed. Einstein's formulation also explained everything that was known about light at that particular time. Moreover, if one counted the number of assumptions in special theory, they turned out to be less than the Lorentz theory. Based on this, Einstein argued that the special theory was superior to the Lorentz theory and later, this was found out to be so (with further experiments).
As we've seen, both the theories explained all the known facts about light. Just because Lorentz's theory required more assumptions than Einstein's, Occam's razor guided us to choose Einstein over Lorentz. Later, as we know, it was found out that special theory was a better alternative for various of reasons. But in 1905, there was no reason other than Occam's razor to choose Einstein over Lorentz.
The above (rather long and boring) story is just one empirical evidence for the validity of the Occam's razor. I find it to be the most compelling one because it was the sole reason to choose one explanation over other. And it proved to be miraculous for the future of physics!