Down on Earth, the shape of the planet is caused by a number of things.
The layer of the Earth that us humans exist on is incredibly thin - the Earth's crust is only 35km thick, only half of a percent of the Earth's total 6371km radius. This thin crust is made up of tectonic plates, and those plates' movement & collision is what creates the mountain ranges, volcanoes, and trenches we are familiar with. When a volcano erupts, the magma cools into intrusive igneous rock while the lava cools into extrusive igneous. These types of rocks then transform into metamorphic and sedimentary as part of the rock cycle.
As a river ages, it goes from a very straight, deep, narrow path to a very shallow, winding one that very easily floods.