Understanding elementary rate laws from a probability stand point
Let's divide the reaction volume into V little boxes and randomly throw ${\rm N}_A$ molecules of A into it and ${\rm N}_B$ molecules of B.  The probability $p_A$ that a given box has a molecule of A in it is ${\rm N}_A$/V, in the limit that $V &gt;&gt; {\rm N}_A$, i.e. an ideal dilute solution, and likewise the probability $p_B$ that it has a molecule of B in it is ${\rm N}_B/V$.  The probability it has two molecules of B in it is ${p_B}^2$, again in the limit that $V &gt;&gt; {\rm N}_B$ and ${\rm N}_B &gt;&gt; 1$.

Consequently, the probability all three atoms are close to one another is $({\rm N}_A/V)({\rm N}_B/V)^2$, which is proportional to the terms in your rate law.  (This isn't a derivation, by the way, but only a rationalization I hope is appealing to intuition.  I've swept all kinds of detail and qualification under the rug.)

Where your intuition may be leading you astray is in the assumption that squaring the molarity of B makes the RHS of your rate law bigger, and that doesn't match up with your intuition that it should be smaller if it involves a less likely event (a trimolecular collision).  But that is not the case.  Squaring numbers only makes them bigger if they are greater than 1, below 1 it makes them smaller.  And of course the prefactor will be different, too.  What the squaring really does is change the dependence of the rate law on changes in B concentration. It says the rate law is much more sensitive to changes in B concentration than A concentration.  And, given that you need two B molecules to collide with each A molecules, that should be intuitively compelling.