What you Need
What to do
Whilst I did this experiment on the show using real urine, I would recommend that you purified something a bit more innocuous, for example salt water. So to start off, dissolve 2-3 teaspoons of salt into about half a litre of water.
Put about 1cm of this salt water into the bottom of your saucepan.
Put the small cup into the saucepan.
Loosely stretch the cling film over the saucepan.
Put a small weight on the cling film so the dip in the cling film is over the cup.
Wait for a sunny day (you may want to do this bit first).
Put the saucepan in the sun and wait for a couple of hours (you may want to tip it slightly towards the sun to heat it better).
Taste the liquid in the cup, is it salty any more?
What may happen
You should find that the cling film rapidly goes misty and then over time the size of these droplets increases until they run down the cling film into the cup.
If you taste them they don't taste of salt (or from personal experience urine - not that I know what urine tastes like...)
Why does it happen?
The pan is a dark colour so when it sits in the sun it will absorb energy and heat up. This heat will be transferred to the water, and the warmer water gets the more water molecules can escape the liquid to form water vapour. However if there are any solids dissolved in the water they won't evaporate so they will be left behind. This means that if the pan is left open, the water vapour can escape so eventually you will be just left with a few crystals of salt in the bottom.
Water that has been evaporated isn't lost completely, if it meets something cold it will condense to form liquid a liquid again. So when you trap the water vapour with the cling film it will condense on the cling film as it is the coldest part of the pan because it clear and not heated by the sun. This means that the water condenses into droplets which grow until the run down the film into the cup. This process is known as distillation and this is a solar still.
So why isn't rain salty?
Similar to your experiment, water evaporates from the sea, and rises, as it rises the air cools, and the water condenses into water droplets in clouds. These water droplets then join together and grow until they fall out of the cloud as rain. As salt doesn't evaporate there is no salt in rain.