Inhale when you lower the weight, exhale when you lift it, and suck your midsection in a bit when you exhale. You actually don't need to flex your abs, just actively suck your midsection in.
The core muscle in question (ask already pointed out by someone else) is the transverse abdominals, not the rectus abdominals that is commonly referred to as the six-pack. Because the modern lifestyle and workplace has people in a seated position almost all the time, the transverse abdominals tends to be any person's weakest point and is a major factor in experiencing back pain.
Anyways, you need to work on sucking your midsection in a bit. Exhaling while pushing/pulling the weights would help a little, because it would provide extra room for you to suck your midsection in (I'll forgo the details).