During lift-off and landing, the methane liquid fuel inside the Starship is being consumed. How constant fuel pressure to the Raptor engines is maintained when the fuel is being depleted and during flip maneuvers?
Each propellant has its own turbopump and preburner. The liquid propellants are pumped into their respective turbo pump/preburners where they are gasified and pass on to the combustion chamber.
To speed the intake of propellants and help prevent cavitation both propellant tanks are pressurized. If nothing further was done as the liquid levels reduced, the pressure in the tanks would slowly decrease as the headspace above the liquids increased.
To prevent this the intention is to use autogenous pressurization. When the Raptor is fired up hot gaseous methane and hot gaseous oxygen are tapped off and fed back into their respective tanks at an appropriate temperature and pressure to maintain the pressure in the tanks. See red and blue up arrows in the diagram
Some of the prototype Starships have used Helium as a pressurant as an interim measure while SpaceX are attempting to solve the sloshing / condensation issue during landing.