The rated TX power is the maximum power emitted while transmitting. The electrical input power is much lower, because the design of the transponder is such that it accumulates energy from the input at relative lower power, to burst that out in one go on the output at high power.
The transmission of a ADS-B extended squitter has a duration of 120 microseconds. Since the modulation is pulse-position modulation, the transmitter actually only transmits for 58 microseconds per message. If the TX power is 250W, a single ADS-B message costs about 0.015 J (assuming 100% efficiency). The maximum squitter rate for ADS-B is 6.2 messages per second, equating to an average of approximately 0.1 Watt of transmitted energy. Other systems hosted by the transponder (Mode S and TCAS) are able to produce easily ten times more messages each. Mode A/C uses much fewer pulses but can have very high reply rates. This brings the average total transmitted power to over 2 W for congested airspace.
Internally in the transponder you will find big capacitor(s) that get charged by the aircraft's electical system, and discharged by the transmitter. They act as a buffer, preventing peak power surges in the aircraft system.