G'Day Alex,
That's all to do with the solar wind... the stream of particles emitted from the sun along with light. At the heliopause, the outward pressure of the solar wind is matched by the pressure of gas in the neighbouring space, so the solar wind basically stops.
The heliopause is egg shaped because of the motion of the sun - less distance to the heliopause looking out "the front" due to "velocity pressure". Not sure if velocity pressure is a scientific term or not, but it's a term we use in engineering to measure the velocity of a gas or air using a pitot tube. The total pressure = static pressure + velocity pressure. So in front of the solar system the total pressure is high (equals static pressure of space + velocity pressure of solar system), to the side it is moderate (total pressure = static pressure of space = static pressure of solar wind i.e. velocity pressure has not effect) and behind the solar system the total pressure is less (equals static pressure of space minus velocity pressure of solar wind - because velocity is a vector!

).
I wasn't aware the heliopause was egg shaped!
Al.