The magnetosphere is important to us, because:
--- It is the environment in which our satellites orbit. Its radiation and outbursts affect unmanned and manned spaceflight.
--- It is an interesting region of space, dominated by electric and magnetic phenomena, not by gravity.
--- It is the most accessible example of "cosmic-scale plasma" which we can study.
--- It has unique links to the solar wind and to the Sun.
--- It is a frontier of science: we still try to understand the polar aurora, magnetic storms and our boundary with the solar wind.
(.. and when we have done that, we still need understand the magnetospheres of Jupiter and other planets, as well as related plasma phenomena on the Sun.)