Why are transmitting antennas generally larger than receiving antennas?
There could be any number of reasons, but here is one.

Better signal to noise ratio.

The signal strength at the receiver is dependent on the product of the antenna gain for both the transmitter and sender.  But the received noise is also proportional to the antenna gain at the receiver.  For the same total antenna gain product, your SNR will be better if the transmitter antenna gain is higher.
Note that Larger antennas will typically have larger antenna gain.  So it makes sense for the transmitter to be the larger one, if one was going to be larger.