It's the magic of constant acceleration. Using the Space Math calculator at
http://www.cthreepo.com/cp_html/math1.htm, if we assume a constant acceleration of 1G for the first half of the trip and constant deceleration of 1G for the second half, it would take about 1.7 days at closest approach (55.75 million km) or 2.85 days at 1AU. Mars hardly moves relative to the Earth in only a few days, so with this kind of speed you can follow a straight line path instead of worrying about messy orbits.
With constant 1G acceleration/deceleration you can visit Jupiter in less than 6 days, Saturn in less than 10, and even Pluto in less than 3 weeks. The 4.22 light year trip to Proxima Centauri would take only 4.8 years (3.5 years to the people onboard, due to time dilation).
Of course, there is a minor tradeoff due to Newton's second law: it takes energy to generate that constant acceleration. In the real world, the Dawn spacecraft's ion drive will produce an acceration a million times less than 1G, and it takes years to get anywhere nearby.
We're still waiting for antimatter drives, negative-mass generator, or the ever-handy space warp.