One should read this quote with the preceding lines in mind, for example, when Beatrice says to Benedick:
I wonder that you will still be talking, Signior
Benedick: nobody marks you.
Courtesy itself must convert to disdain, if you come
in her presence.
Beatrice expresses disdain for Benedick, and the words
I had rather hear my dog bark at a crow than a man swears he loves me
are a continuation in the same vein.
Human speech, unlike the barking of a dog to a crow, has meaning to humans; Benedick's words have as little meaning to her as a dog's barking.
One should not too quickly jump to the conclusion that she considers Benedick's words as "annoying noise" (see Mithrandir's answer). It was perfectly possible for Elizabethans to consider the sound of barking as pleasant. Witness the following passage from Act IV, Scene 1 in A Midsummer Night's Dream, where Hyppolita talks about a hunting experience (emphasis mine):
I was with Hercules and Cadmus once,
When in a wood of Crete they bay'd the bear
With hounds of Sparta: never did I hear
Such gallant chiding: for, besides the groves,
The skies, the fountains, every region near
Seem'd all one mutual cry: I never heard
So musical a discord, such sweet thunder.
However, in this case, the hounds were chosen in such a way that their combined barking is somehow "harmonious". (I think there is at least one other play where Shakespeare mentions this sort of thing.) The barking of a single dog may still sound monotonous though.
So my interpretation of Beatrice's words is that she finds Benedick's talk both less meaningful and more monotonous than the barking of a dog.