Russell explains Ockham's Razor (also known as 'Occam's Razor') as follows in 'History of Western Philosophy':

"This maxim says: 'Entities are not to be multiplied without necessity'. Although he [Ockham] did not say this, he said something which has much the same effect: 'It is vain to do with more than can be done with fewer'. That is to say, if everything in some science can be interpreted without assuming this or that hypothetical entity, there is no ground for assuming it. I have myself found this a most fruitful principle in logical analysis." (p.462-3)

It is interesting to note three facts here.

First, the principle may be most useful in logical analysis, in metaphysics, and elsewhere, but it is not itself a principle that can be deduced from mere logic.

Second, what amounts to the principle may be supported and explained by probability theory, for there we have the principle that p(P&Q) <= p(P), that is: A conjunction of statements - such as assumptions - is never more probable than any of its conjuncts.

This still does not imply Ockham's Razor, but it does imply it if we make another assumption

It is desirable to choose the most probable explanation

which can be defended by the consideration that if one does not choose the most probable explanation, but one that is less probable, one has - to the best of one's knowledge - more chance of being mistaken. And with reference to what Ockham did say i.e. 'It is vain to do with more than can be done with fewer', it may be noted that every further assumption involves a further chance of contradicting the facts.

Incidentally, Ockham's Razor is also a principle that suggests that one acts vainly if one assumes divinities. See Religion.

Third, Ockham's Razor was elevated to the fundamental Rule of Reasoning that Newton added to the 2nd edition of his Principia

Rule I : We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

which, in turn, may be seen as having caused (!) Hume's doubts about induction. See: Newton's Rules of Reasoning.