It is more green. Houses have to breathe. As people have mentioned if the windows aren’t opened and the air exchanged inside the house regularly the air goes stale, smells bad, and grows damp. This causes all sorts of problems- condensation and mould being the biggies.
Normally houses leak air- cracks in the floor, poorly fitting windows, up chimney breasts or whatever. Houses are constantly leaking air: this is felt as a cold draught. While leaking air, houses are also leaking heat. Thus heating systems have to work harder and harder to compensate for the incoming air (which in winter is very cold).
By closing down all air leaks and making the house relatively airtight, those heat leaks are lost. Then by putting a heat exchange on the air pumping system, about 70% (I believe) of the heat is actually retained.
If you have modern walls that are insulated to a high degree (no gaps, around 1-foot of solid PIR insulation, and very carefully closed up around windows/doors) you can achieve “passivhaus” standards. The thermal mass in your walls will compensate for the heat loss in the exchanger. I’ve heard of people who spent a tenner a month on heating in winter. I did also read a post about someone who said mostly they only turn the heating on when their kids aren’t there because without the thermal output of the children the house can grow chilly overnight.
It’s a brilliant system- I wish I had it. If you’re getting a new build I seem to remember the cost being around £10-£15k, and most of that was the heat exchanger itself. It’s apparently not that complex to install (although the design has to be right or you end up with rooms that are sluggish or stale which is bad). OP could probably tell you better about the cost as well. Also she has a water heat exchanger which I’ve not heard of but am off to google once the baby is in bed.