What does this actually mean and why did some Ancient Greek philosophers seem to believe that the heart was the seat of the soul?
See Michael Davis, The Soul of the Greeks: An Inquiry Chicago UP (2011): in Homer, psuchē is not yet “soul”; it means something like “the breath of life.” Homer does use a collection of words to refer to what one might call our inner life, depending on the context, they may mean “spirit,” “breast,” “mind,” “heart,” or “the seat of our life”: what animates us.
And see Greek's concept of the soul:
Given the idea that soul is the distinguishing mark of all living things, including plants, the Greek notion of soul is, as we have seen already, broader than our concept of mind. For it is at least conceivable, and probably true as a matter of fact, that there are living (hence ensouled) organisms without minds, without, that is to say, desire and cognition by sense or intellect.
In another way, the conception of soul that is in evidence in Plato's Phaedo is significantly narrower than our concept of mind, in that the soul, as conceived of in this particular dialogue, is not, in fact, responsible, or directly responsible, for all of a person's mental or psychological activities and responses, but only for a rather severely limited subset of them. Socrates attributes a large variety of mental states not to the soul, but to the (animate) body, such as, for instance, beliefs and pleasures, and desires and fears. At the same time, the soul is not narrowly intellectual: it too has desires, even passionate ones, and pleasures as well, such as the pleasures of learning.
Aristotle consider three progressively more complex versions of soul, each incorporating what comes before it. He first treats soul as nutritive (plant soul), then as perceptive (animal soul), and finally as cognitive (human soul).
Aristotle devotes most of his energy in De Anima to detailed investigations of the soul’s individual capacities or faculties, which he first lists as nutrition, perception, and mind, with perception receiving the lion’s share of attention. He later also introduces desire, evidently as a discrete faculty on par with those initially introduced.