FPGA designs are based on lookup tables (and a few other basic elements). The lookup tables are small RAMs, and can be configured to perform a specific logic function after the device has powered up. Connections to these LUTs connects to various muxing logic, which can also be configured after the device has powered up.
Placement is selecting which LUT on the FPGA will perform a specific logic function. Routing is configuring the muxes so the LUTs take their input from the correct source.