One more concept to realize is that since antennas are passive devices, they have to be reciprocal. In other words, radiation that goes through a reciprocal system has to work the same way forwards and backwards. In simpler words, if you imagine transmitting through the antenna and looking at the radiation pattern from far away--that antenna behaves exactly the same when the radiation shows up from far away receiving into the antenna--it behaves the same in transmission and reception.

Each phased array antenna element can be thought of as a radiator, and if its an active array, the radiating element can be thought of having a tuneable amplitude and phase. And the combined radiation pattern of the array is a superposition of all the individual elements.

When you have that kind of control, you can change the far-field radiation pattern by adding delays and changing the amplitude to tune the radiation pattern. It's quite ingenious actually.