What would be a reasonable scientific way of explaining a railways-like travel system between planets of the same solar system?
Interplanetary Trains Running on Tracks? No!
There is the concept of a beanstalk space elevator to save fuel on the way to orbit. This requires materials science beyond what we can do now. Some visionaries wave their hands and say "nanomaterials," but we're not even close yet. A similar system on an interplanetary scale would need more and better materials.
Interplanetary Craft Running on Train-Like Schedules? Yes!
Consider a cycler spacecraft running between planets, and orbital transfer stations to collect the payload in orbit. That would be much like trains and stations, but there would be no platform where the passengers assemble. In effect the train doesn't stop, people and cargo have to catch an orbital transfer vehicle instead. Still, there would be the "4.50 from Paddington" and if you miss it you have to wait for the next one.
Perhaps a better way to think of the look-and-feel would be ocean liners on a schedule. You board it, perhaps from lighters, then it goes to the next port.
Perhaps not fully scientific yet; they're theoretical in nature and modern theories say they would be small and unstable; but given that this is future state Sci-Fi, one could argue that a system had been found to create large, stable wormholes between specific locations within a solar system.
The real issue is accounting for planetary orbits. That kind of fits in though because given their inherent theoretical instability, they could be allowed to collapse on a regular basis and then recreated in a more closely aligned position on (say) a daily basis or so.