Hi!
Look in the September 1976 Issue of the UK's "Practical Television" magazine (free from American Radio History site) and there you'll find construction plans for exactly the sort of instrument you've shown, based on a 555 timer chip!
The capacitor is periodically charged and discharged by the 555 and by integrating the area under the voltage – pulses with respect to time over the pulse duration (Vc = 1/C x integral v.dt) the meter will read the average current, which can be shown to be directly proportional to the value of C – a short appendix is provided showing the formulae!
Chris Willlams