One of the main reasons that slow practicing is so effective is that it allows us to fix problems much more easily than when we play at full tempo. Passages that are difficult to put together can often slide into place when slowed down. Our hearing of detail can be boosted considerably, and we can hear things that eluded us before.
Another element that I emphasize during slow practice is to suspend judgment regarding our personal opinions of our playing. Take yourself out of the picture. Give yourself permission to fail and proceed with a sense of gentleness and objectivity. It's okay if you can't play a certain passage yet--something isn't yet in place and the slow approach is a great way to find it.
Mildred Portney Chase in Just Being at the Piano (my all-time favorite book on piano playing) wrote a wonderful 4-page chapter on the journey of slow practicing. Here are two noteworthy quotes:
One final thought--what keeps people coming back to the joys and tribulations of practice year after year is that "Aha!" moment, where our understanding of our instrument, our body, or the music suddenly springs into focus and we get that inner feedback that moves us onward to the next challenge. Slow practice is one of the tools that can get you there.
By practicing a small segment a few times, you may realize greater improvement than in many repetitions of the whole phrase. it is possible to jam the mind's programming by presenting it with more than one problem at a time. One inch on the page may require more difficult adjustments than twelve inches somewhere else. It is important to know how much you are really understanding. However much time your mind needs to absorb the material, that is how much time to allow yourself. Show yourself the same kind of patience that you would a good friend who could not move through the material as fast as you would wish...
...Slow practice allows your knowledge to be integrated with your playing, allowing thoughts to become feeling. It removes the interference that comes from trying to think movements into place. I like to think that all knowledge should float freely into place, finally settling as though it were a mantle of snowflakes, so light as to fit into the nooks and crannies of oneself. Slow practice is a setting in which this can take place.
Next: 5 Reasons to Memorize Music