The most popular formulation of the principle says that you just have to seek for the simplest explanation.
“We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.” — Newton
But there is a deeper formulation, that concerns the very nature of explanations, hinted here in Wittgenstein’s Tractatus:
3.328 “If a sign is not necessary then it is meaningless. That is the meaning of Occam’s Razor.”
In a logically correct explanation of a phenomena, you are not allowed to invoke any entity without using some part of the phenomena as a justification for your claim.
E.g. if your apartment has been robbed, you are allowed to say stuff like:
1) “There is a rope hanging from my window, therefore the robber used it to enter”
or even less probable stuff like:
2) “They didn’t steel my shampoo, therefore they must be bald”
3) “The thief is probably a person of color”
The difference is that the claim 3) that the thief is a person of color, has no basis, whereas there is some basis, for claiming 2) that they are bald, (although not much.)
Statistically, 3) might be probable, more probable than 2), but still 3) is forbidden and 2) is allowed, cause when we make claim 2) we are thinking, whereas when we make claim 3) we are just repeating stuff that we heard before.
The scholastics (just like the Greeks) had faith that God left clues that would allow them to discover the truth about the world by themselves. This is the basis for Occam’s razor - faith in one’s own abilities. And this is what makes it important.
Also, one might consider the principle that is dual to Occam’s razor:
Occam’s razor (or Negative Occam’s razor, as I sometimes think about it): Entities must not be multiplied beyond necessity i.e. when forming a theory, postulate no more than it is needed to explain the phenomena.
Dual Occam’s razor (or Positive Occam’s razor): Entities must not be removed beyond necessity i.e. if something is indeed needed to better explain the phenomena, it must be postulated.
David Deutsch uses this to makes a very compelling argument in “The fabric of reality”, regarding the Many-worlds hypothesis of quantum mechanics that goes roughly like this: “if the other universes don’t exist, then where does the computation of quantum computing algorithms, such as Shor’s alghorithm, takes place?”