Stabilizing is causing something to become stable. If your presence is stabilizing, or even settling, to a person, you are probably a good influence on them.
You probably realize that the root of this word is stable, which means, "not likely to change or fail." So when engineers design vehicles that transport us or buildings that we live and work in, they integrate all kinds of stabilizing elements for safety. One such feature, on the wings of airplanes, are flaps that the pilot can control during flight to adjust speed and angle of descent.