There is still no categorical answer as to why humans, and no other species, have speech, or why speech is the way it is. Several purely anatomical arguments have been put forward, but they have been shown to be false, biologically implausible, or of limited scope. This perspective paper supports the idea that evolutionary theories of speech could benefit from a focus on the cognitive mechanisms that make speech possible, for which antecedents in evolutionary history and brain correlates can be found. This type of approach is part of a very recent but rapidly growing trend that has already provided crucial insights on the nature of human speech by focusing on the biological bases of vocal learning. Here we contend that a general mechanism of attention, which manifests itself not only in the visual but also in the auditory modality, might be one of the key ingredients of human speech, in addition to the mechanisms underlying vocal learning, and the pairing of facial gestures with vocalic units.