There is no true edge to the atmosphere, or a beginning to 'outer space'. The atmosphere just gets thinner and thinner. Even at altitudes of 10,000 kilometers - high above the orbit of the Space Station - there is still a lot of atmosphere left. The contact point or 'interface' between the magnetospheric system and the atmosphere reaches deeply into the atmosphere and ends at about the location of the ionosphere some 100 kilometers above the ground.
The ionosphere is an electrically-charged layer of gas that we use to reflect radio waves between distant stations. Energy and currents flow from the magnetosphere into the ionosphere, and back out into the magnetosphere, like an invisible electrical current. Along the way, some of these currents can cause the Northern and Southern Lights (Aurora Borealis and Aurora Australis) in the polar regions of Earth.
Meanwhile, lightning storms in the lower atmosphere cause electrical currents to flow into the upper atmosphere and ionosphere. The ionosphere receives energy both from the magnetosphere and from electrical storms in the lower atmosphere. This also means that the magnetospheric system can have some affect upon atmospheric electrical systems near the ground and in the lower atmosphere (troposphere). The sun also causes changes in the atmosphere above the ozone layer, as its ultraviolet light is absorbed and delivers energy to atoms there.