What are some possible ways this could happen?
Possible? yes, though highly unlikely. Much more plausible would be a system that only has gas giants that have ejected the other planets.
Way back when: Jupiter and Saturn had harmonically resonant orbits and this is surmised to have helped eject all the other garbage out of our system (anything without a stable orbit got pushed and pulled and tossed out). Without gas giants to protect your system, you're not going to be able to charge good rent money.
Something similar might have happened in your system, but instead the gas giants ejected themselves. Note, this still leaves yours in 'the cheap seats' but gets around the unlikely event of a system having had to form without gas giants.
I've nothing to back this up, but I'd assume that's why most stars are binary: you either don't have enough 'stuff' to start a star at all, so much that you get two stars, or like ours: gas giants as well.
There are a number of presentations on planet formation in the SETI Weekly Colloquium Series covering the latest understanding and speculation on the subject.
I suggest you start with one of the formation models presented, well-peppered with real nomenclature and details, and also cite the specific mechanisms and situations for getting rid of the other planets and leaving the configuration you want. That is, mix and match among the real mechanisms to craft the scenario you want.
It depends on the size and composition of the proplanetary disc . If it is fairly small , with a low amount of hydrogen and helium , then I'd say yes , it would be possible , even with a large proplanetary disk , if the percent composed of hydrogen is low enough , than yes , you could get a solar system with no gas giants , but a side affect may be a vastly larger amount of terrestrial planets and asteroids
A star is made of gas. Some of it ends up in the planets, it would be awfully hard to make a solar system without gas giants. However, something could have happened to them. While I can't see a mutual ejection scenario working (where does the energy come from??) you could have an ejection/destruction scenario (one planet leaves, the other ends up too close to the central star) or ejection by an encounter with an outside body passing through.