Imagine that your car begins making a strange knocking sound when you brake. You take your car into the autoshop to find out what's causing the problem. Without even looking under the hood, the mechanic explains that the reason your car is making the sound is because there are infinitely many universes with different versions of you and your car, and you just happen to live in the universe where the car began making a knocking sound. Odds are, you'd look for another mechanic.
The multiverse theory essentially uses the same logic to explain (or explain away) features of the universe such as the gravitation or planck constant. As such, its basically a huge non-explanation.
Regardless of Davies particular bias, his main point is that the theory is practically devoid of explanation and is about as meaningful as saying "God did it". When Davies invokes the term "falsifiable", he isn't claiming that there couldn't be evidence to support it, but rather there could never be a way to disprove the theory. The lack of criteria for falsifying multiverse theories is problematic because there could be other, more fruitful theories that explain the same observations as a particular version such as Brane Theory.
Returning the above example with the mechanice, compare the two rivals theories for the noise:
a) infintely many universes with different parameters for the car's noise level when brakes are applied.
b) the brake-pads are worn out.
A is just as valid a theory as B; the difference is, you can disprove B, and apply B to tell you something about the car (it needs new brake pads).
Which theory do you think is better?
For a final bit of flame; yeah, maybe people dislike multiverse theories because they mean are universe isn't "special"...but my feeling is that multiverse theories have such appeal is because they're a neat sci-fi prop, not because they have any real value.