Lithography is a kind of printing process which is used to create the circuitry on the chip. It's a bit like taking and developing a (non-digital) photograph.
The die size is the physical size of the actual silicon chip (which is rather small in relation to the "package size" of the whole processor).
Nanometres usually refers to the production process used to create the circuitry. The more nanometres, the larger the features on the silicon are, and the less circuitry can fit on the die. The smaller the number of nanometres, the more complex the circuitry can be.
I'm not sure exactly what is measured to arrive at the number of nanometres, but it does relate indirectly to the minimum size of a detail within the circuitry on the die. This number is not in itself the actual size of the smallest possible detail. However, the smaller the number is, the smaller the smallest possible detail is.
The physical size of the processor die is limited by the speed of light; if you make it too big, the electricity takes too long to move around the circuits and so you have to reduce the clock frequency of the processor to make sure all the electricity gets to where it is supposed to go before the next clock cycle starts.
You can't increase the speed of light, so if you want the electricity to finish up where it needs to finish up in less time than before, you have to shorten the distance it needs to travel.
Therefore, if you want to add more circuitry to allow you to do more calculations or actions in each fixed clock cycle, you have to reduce the physical size of the entire contents of the chip so that you are not adding too much time to the journey of the electricity through the circuits.
Likewise, if you require a faster clock speed, you have to make the entire contents of the chip smaller so the electricity can travel around the same circuit in a shorter time.