Why is the shape and size of Earth's magnetosphere different on both sides?
The structure and behavior of magnetospheres is dependent on several variables: the type of astronomical object, the nature of sources of plasma and momentum, the period of the object's spin, the nature of the axis on which the object spins, the axis of the magnetic dipole, and the magnitude and direction of the velocity of the flow of solar wind.
The distance at which a planet can withstand the solar wind pressure is called the Chapman–Ferraro distance.
Over the Earth's equator, the magnetic field lines become almost horizontal, then return to connect back again at high latitudes. However, at high altitudes, the magnetic field is significantly distorted by the solar wind and its solar magnetic field. On the dayside of the Earth, the magnetic field is significantly compressed by the solar wind to a distance of approximately 65,000 kilometers (40,000 mi).
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In short: The Magnetosphere of a planet has its asymetrical shape due to the Magnetic pressure of the solar wind.
On the sun side of the planet the solar wind squashes the Magnetosphere untill the magnetic pressure of the solar wind is equal to that of the Magnetic pressure put out by the planet. On the night side of the planet the magnetosphere is under a lot less pressure from the solar wind due to the protection of the planet, but still comes under pressure from the "sides"as the solar wind "expands" causing a very long tail like effect.
The simplest analagy that people will be able to relate to is a rain drop. As the rain drop (planet + magnetosphere) falls to the ground (sun) the pressure of the atmosphere (solar wind) blunts the ground side, and the skyward side is extended through lack of resistance. The magnetosphere goes through much the same forces, only the ground and rain drops would be still and the atmosphere would be constantly rushing up.
The Funnel like cusps you see are something a little more complicated arrising from the shape of a magnetic field coming out of its source, and a phenominon known as magnetic recconection (and is one of the causes of Aurorae).