About the cathode ray tube used...
It uses electrostatic deflection of the electron beam, which passes between the deflection plates. Normal there are 2 x 'X' plates for horizontal deflection and 2 x 'Y' plates for vertical deflection. The voltage is applied across them, moves (deflects) the electron beam, proportional to the voltage. There is no connection between the plates except a small capacitance. The parts are all in a high vacuum, and the only connection that could cause current is if the electron beam hits one of the plates. There would be a small current whenever the voltage changes due to the capacitance. Note the electron beam is probably only a mm or so diameter. The deflection plates are probably 5 or 10mm either side of it.
http://www.circuitstoday.com/crt-cathode-ray-tube
In practice driving the deflection plates directly is very inconvenient, because the voltage for deflection is significant, and the sensitivity and position need to be adjusted to understand the resulting deflection. Additional components and bias voltages are required. Note the voltage is differential, the difference between the opposite pair of plates. Modern oscilloscopes do not have a direct external connection for the deflection plates.
An amplifier is used to drive the deflection plates, with the various positioning voltages added. The gain is known and can be adjusted so the deflection is proportional to voltage. The signal input of the amplifier is usually reference to ground on one side, and the input resistance is usually one megohm. This resistance does not change at different gain settings. A "probe" is used to connect to the circuit, which is designed to work over a broad frequency range like the oscilloscope itself. For most situations 1 megohm is likely to change the voltage being measured, so a divide by 10 passive probe is used for most measurements, that makes the input resistance 10 megohms, but reduces the voltage by 10. This is usually sufficiently accurate. More importantly the probe reduces the input capacitance by 10 too, so that the disturbance to the electronic circuit being observed is minimized. The input capacitance might be 20-100pF, depending on just how well designed the circuitry is. With the divide by 10 probe it is then 2-10pF across the circuit being measured, usually acceptable. Note that the reactance of a capacitor of 10pF at 20MHz is only 796 ohms. There are active probes with much less capacitance (0.2pF) but these are quite expensive and have a limited voltage range.
This link explains all about passive probes. Note there is a compensation adjustment, which is set correctly using a square wave signal provided by the oscilloscope for the purpose.
http://www.google.com.au/url?sa=t&rct=j&q=&esrc=s&source=web&cd=7&ved=0CGkQFjAG&url=http%3A%2F%2Fcircuitslab.case.edu%2Fmanuals%2FProbe_Fundamentals-_Tektronix.pdf&ei=RQnmU_7jGIve8AWu1YDgDg&usg=AFQjCNHgJaRqGDaezXk0A-2TqybhN8XLeg&sig2=7pPL0y13rVWI40Fu-4z30A&bvm=bv.72676100,d.cGU
The advantage compared to a voltmeter is that the waveform can be shown. A modern voltmeter can have a much higher input resistance, can be connected across two points in a circuit, and is probably more accurate, but the oscilloscope can measure much higher frequencies, show the shape of a waveform, and observe events. Note that most modern oscilloscopes do not use a CRT any longer. They are called digital oscilloscopes, and are wholly electronic, with a computer type display. However the input circuit and even the probes are very similar.