There are many ways that a voltmeter works. Unfortunately because of the variety of different methods used I cannot say how the one you have works. Also this is such a basic instrumentation question that I wonder how I can explain any of the more accurate methods. This is like asking how does a spring work?
Well come to think of it a spring is a pivotal part of exactly how an analog meter does work. First thing though let me talk about a spring. A spring is a linear device that generates a force in the opposite direction that one end has been moved if the other end is fixed in place. To stretch a spring a distance X the force needed is F. To stretch the spring a distance 2X requires a force of 2F. Now the force generated by an electromagnet on a permanent magnet is proportional to the voltage applied to the electromagnet. When the electromagnet pulls a magnet that pulls a spring we can measure how far the one end of the springs movement to find out how much voltage was applied to the electro-magnet.
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"Don't disturb my circles." translation of Archimedes last words
A VMM (or DMM) generally consists of a voltage divider to scale down high voltage to a more manageable level for use with internal IC's. Now when the probes are connected in parallel to your circuit, that voltage divider shares the same voltage of that part of the circuit due to the laws of resistors in parallel. Now that the DMM knows the ratio of the divider within, it uses IC's to calculate and display the information. As for settings I'm not sure what exactly you are asking for, but if you are asking about range, that is simply a switching of the ratio for the voltage divider to bring the voltage that will be fed to the IC's to an appropriate level. As for current there is a similar event happeneing. When you connect to your circuit, you connect it in series so that the same current in that part of your circuit flows through the same voltage divider (which is just a couple of resistors in series). So since the DMM know's the resistance that the current is flowing through within the DMM it can calculate the current based off of the voltage created in the divider.
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The early bird may get the worm, but the second mouse get's the cheese.