The datasheet says 1KHz square wave with a 6V amplitude approximately. I'm going to guess that 6V is peak to peak as the thing runs of a 9V battery.
So grab a 555, select the resistor capacitor combination(plenty of online calculators) to give a 1KHz signal, add some 9V battery leads, put a big capacitor in series with the output, and tada. You have your signal generator.
As for the probe, the nail with wire wound around it sounds like the thing to do. This is just a simple transformer, a terrible, but simple transformer. You would have to experiment with the number of turns. After you have your nail transformer you will need to amplify the signal. A little head phone amplifier might be enough, like the LM386.
And for any one else here is the datasheet of the tester from amazon: http://www.extech.com/instruments/resou ... 180_um.pdf