Different stations are usually well conneted by road, light-rail, subways, express buses and taxis.
The reason is quite simple why many cities have multiple stations. Before the age of HSR most cities had 1 or 2 rail stations. For example in my city Zhengzhou that would be the Zhengzhou Station. It is a major rail transit hub and located right in the city centre. However when HSR is built there came a problem, the station needs to expand to accommodate dedicated HSR railways but it's already at its limit when it comes to room for expansion as the surrounding land have all been developed. It would be extremely costly to tear down nearby builidings just to expand the railway station.
There was another prevalent problem, which was that the city itself was cramped and very crowded. It needs to grow and expand just like the station. So the solution to these two problems became rather obvious, to build a brand new, dedicated HSR station, far away from the city centre but located in an area close to the city that future development projects are concentrated in. Think of this not as building a second train station for one city, but a new station for a new city right beside the old city.
This solution solves multiple problems at once. First, design can start from scratch to make sure it is most state-of-the-art. Second, it allows HSR lines themselves to be off-centre as well, hence making land requisition much simpler, faster, and cheaper. Third, it provides a transport hub for the new development centre for the city. Fourth, it helps to feed growth to the new city centre, which in turn accelerates the new station's development. Fifth, it stimulates construction of connecting infrastructure, such as roads, light-rail, and subways. Sixth, it spreads out real-estate development as well, because new developments usually chase after such stations and along connecting subway stations. If the city is too concentrated, so will property prices. If transport hubs are more dispersed, then property prices will also be more evenly spread out.
In fact this model was so successful, that during the later stages of the HSR's development in China, many cities started to deliberately locate the new station to be off-centre, because the growth that it spurs is just enormous wherever it's located.
When this process repeats itself for nearby cities, what happens is that these cities, with their original city centres far away from each other, will gradually fuse together and become a continuous city group. This is most evident for cities in the Yangtze River delta (around Shanghai) and cities in the Pearl River delta (around Guangzhou, Shenzhen).