DIAGNOSIS That's the classic symptom of a clogged or misadjusted carburetor, usually because old fuel gummed up the tiny internal jets. The saw gets enough fuel to start but then starves.
FIX First, drain all the old fuel and mix a fresh batch of 50:1 gasoline and Stihl 2-cycle oil. If it still dies, you'll need to clean the carb. Remove the air filter cover and filter, then the carb cover plate. You'll see two adjustment screws: the high-speed screw (H) and the low-speed screw (L). Gently turn both screws clockwise until they lightly seat, then turn each counter-clockwise one full turn out. This is a baseline setting. Reassemble and try starting.
IF THAT DIDN'T WORK The fuel filter inside the tank might be completely clogged or the fuel line might have cracked. You'll need to pull the tank apart to check. If you see a little white plastic filter that's brown and gummy, that's your culprit.
HEADS UP Don't just crank the adjustment screws wildly—they're very sensitive. A quarter-turn makes a big difference. If cleaning doesn't work, the carb diaphragm might be stiff and the whole carb might need a replacement kit.