how much more: This is an argument from lesser to greater, a traditional Jewish way of reasoning. The idea is that if something is true in a smaller case, it is even more true in a bigger case (see also Hebrews 2:1–4; 12:25–29). If animal blood had some effect in cleansing, then Christ's blood will be much more effective.
purify our consciences: Christ’s sacrifice, unlike the old covenant sacrifices, removes the guilt that keeps us from God by thoroughly cleansing us from sinful actions.