Why does the relay in this touch circuit not work?
This doesn't work because the current thru the touch sensor is small and the gain of the transistor isn't large enough to amplify it to what the relay needs.

You haven't provided any specs on the relay, so I'll just make up numbers for sake of example.  Let's say the relay needs 30 mA at 9 V to operate.  You haven't provided a link to the datasheet of the transistor, so let's say it's guaranteed minimum gain is 50.  That means the minimum base current to operate the relay would be (30 mA)/50 = 600 &micro;A.  It is quite unlikely that this touch sensor will let 600 &micro;A pass.

Let's work backwards to see what the resistance between the touch sensor terminals would need to be.  Figure 700 mV for the B-E drop of the transistor.  That leaves 8.3 V across the touch sensor.  By Ohm's law, (8.3 V)/(600 &micro;A) = 13.8 k&Omega;.  That's low for ordinary skin, but could probably be achieved by wetting the skin with salt water first.

However, beware of the current thru the body.  If this touch sensor is passing current between different parts of the same finger, then you might only feel a tingle.  If this touch sensor connects two fingers on opposite hands, then this would actually be quite dangerous.  Nearly a milliamp flowing near the heart is a bad idea.

In any case, the solution is more gain.  A second transistor could be used.  That would also allow a resistor in series with the touch sensor to guarantee the current thru the body would be limited.  This is something your circuit is NOT doing now.