What does the following sentence mean matter recycles again and again but the energy passes through the ecosystem only once

If my understanding is correct, what this saying is referring to is the conversion of energy from relatively useful forms to heat energy (which is relatively useless).
As you probably know, conservation of matter says that matter can be neither created nor destroyed. You may also be aware of the conservation of energy, which states that energy can be neither created nor destroyed, but only converted from one form to another. Whenever you have a chemical process, a complicated process called entropy is what drives the process (you don't need to know exactly what this is, but if you want to, you can look up the second law of thermodynamics). Part of that process often involves energy being converted to different forms--from useful forms (that can do work) to useless forms (that can't do work). Useful forms of energy are "useful" because they can do work, and in the process are converted to heat, which is not very useful for work, and generally cannot be converted back to useful forms. Matter, on the other hand, does not have "useful" and "useless" forms.