Why does your lawn mower start then die?

Two reasons: Either fuel starvation or flooding. If you are getting black smoke when the engine starts, it is being flooded - check the spark plug and the air filter and clean or replace, check the carburetor and choke adjustments and reset as necessary. If the engine is not blowing black smoke, it is starving for fuel - look for a dirty/clogged fuel filter; a kink in the fuel delivery hose; carburetor adjustment for a slightly richer mixture.