There are a few factors that make this possible:

1. As others have pointed out, the link budget (how much energy loss a particular radio link can handle before it is broken) for D2C satellites assumes a nearly direct line of sight from your handset to the satellite. This is much easier to achieve with satellites in space than it is with traditional cell towers that might have numerous walls/buildings in the way.

2. The D2C satellites use massive phased array antennas that are able to point a very narrow beam very accurately to the ground. This provides a substantial amount of antenna gain that further helps the link budget. The gain from the antennas allows the satellites to pick up even relatively weak signals from a handset.

There are other tricks as well, but these account for the largest differences. Of course, doppler gets in the way, but it is a solvable problem.

My thinking is that you can think of Starlink satellites as LTE towers that just happen to be ~350 miles away from your phone. It happens to work because while they are far away, the satellites have a very clear line of sight (directly down) with few (no) obstacles.

The complication is that the base stations will be moving much more rapidly than traditional terrestrial towers.

very slowly, with the radio at maximum power, and no obstructions

Your cell phone uses the nearest base station, the base station will handover to the satellite infra.