First you get a basic 315MHz transmitter and receiver, ensuring that the transmitter can be powered from a small 9V battery (for simplicity I have assumed the receiver can also be powered from a 9V battery and that no tuning is required).
Generally speaking, when power is applied to a transmitter, it begins to transmit an unmodulated carrier. This will result in the receiver (if within range) having an output that takes a DC value (it may be high or low, depending on the receiver).
Assuming the receiver's output is digital (or DC coupled) you can connect a LED directly up to the output (perhaps with a current limiting resistor -- not shown) and it will either turn on or turn off when the carrier is received. You can play about with the connection of the LED and you may be able to get it to turn ON when the carrier is received.
At this point your circuit does what you basically want.
However there are some issues:
1) another transmitter on the same frequency will also turn the LED on.
2) with your transmitter off, the LED may remain ON, or ON at a reduced brightness, or it may flicker on and off (or it may just turn off)
3) If the battery on the receiver goes flat, then the receiver may fail to operate, it may operate with reduced range, or the LED may simply be less bright.
4) If the battery on the transmitter gets flat, it may stop operating, or operate with reduced range.
5) If the transmitter is moved with respect to the receiver, the LED on state may be determined by obstacles, state of charge of the batteries, and distance.
In some respects this solution satisfies your requirements, since you do not actually specify what is to happen when the battery is disconnected from the transmitter (actually, a LED connected to a battery via a resistor technically satisfies your requirement of a LED being on while another battery is connected to a transmitter)
However, I suspect you want the LED to go off when the battery is removed from the transmitter
Here is where things can go wrong as I have suggested above. Because the receiver is still active, it will try to pick up a signal an many receivers will just output noise (like the hiss you hear between stations on a radio).
For this reason, a practical device will typically transmit a modulated signal, and the receiver will look for this modulation to signify that it is (or is not) receiving a valid signal. This adds extra complexity, but pretty much fixes points (1) and (2) above. The other three points are practical limitations based on availability of power (or transmit power/receiver sensitivity).
