For Aristotle, every natural kind has an essence. In this respect, his view has some similarity to Plato's account of the Forms. The difference is where the essence/Form is. For Plato, the essence is in our souls / in the mind of God. For Aristotle, We intuit this essence when we look at the thing, and we learn the essence in this way.
Aristotle uses the term "soul" to refer to the essences of living things. These essences have activity -- i.e. life. Thus, it is the nature of plants to require sustenance and grow and reproduce while they are alive. If you kill them, then what remains in Aristotle's view is not a plant. Animals join to this further abilities of sensation, motion, and to some extent cognition. It is their nature to do these things and thus part of their essence / soul.
Thus, the quoted passage is stating that you cannot speak of the bodily parts of a living thing without simultaneously referring to its soul. Part of the reason is that the soul is what we intuit that makes the thing that kind of a thing. In other words, if I look at a dog's tongue, when I recognize it is a dog's tongue, I am referring not just to the arrangement of atoms that are the dog's tongue but also the fact that it is arranged, living, and functioning as dog's tongue. In Aristotle's vocabulary, it is actualized as a dog's tongue. The activity is that of soul as organizing matter dogwise and then as a feature of this toungewise. The thing acted upon is matter (or in Aristotle's vocabulary passive). Thus, he's saying it makes no sense to speak of the matter of an animal, plant, or human without simultaneously speaking of its soul.
Part of the reason goes back to the first difference I highlighted between Plato and Aristotle. For Plato, when we look at things in the world and see them as things, we are looking at a bad shadow / imitation of the Form. For Aristotle, we are looking at matter and recognizing through our senses (through the creation of an image [lit. phantasm] and then the use of our mind an essence in the thing. Thus, we see something that is formed matter as we see the activity [soul] arranging the matter according to the kind.