What Makes a Radio Transmitter Powerful?
Very simply, transmitter is "more powerful" if it puts out more power.  It uses a higher voltage and higher current from its "more powerful" power source, and it outputs higher voltage into a given antenna impedance (typically 50 ohms).

So, a transmitter that outputs 7V into the 50 ohm antenna is putting out 1 Watt. And if the transmitter is designed to output 225V into the 50 ohm antenna, then it is outputting 1000 Watts.

There is also some amount of gain you can get from the antenna itself.  Some antennas can make the ERP (Effective Radiated Power) 2x or more what the transmitter is sending up the feedline.