TL;DR: Grounding is not necessarily important for touchscreens to work. However it influences some sensors.
I think there are some basics required to answer this completely. Obviously a capacitive touchscreen has something to do with capacitance. It recognizes objects, by the change of capacitance they facilitate. Now the interesting bit is: What capacitance exactly does change and how is this capacitance measured by the device?
There are different takes at this. I'm going to go with the one I know best: mutual projected capacitance, which is also the method used in the iPhone touchscreen.
This type of sensor uses a matrix of conductors which criss cross the transparent surface (usually glass with etched and printed surface) and thus form crossings. At these crossings no electrical connection in the sense of low resistance is formed but the conductors are very close to each other and thus form the plates of a capacitor with the dielectric between and around them. In this regard the "around them" is also the air in front of the glass.
When your finger is in front of the glass it thus replaces a part of the dielectric of these small capacitors. Your finger and also wool has a different dielectric constant (which is just a comparison of how strong the charges in the material react to outside electrical fields) than air. The capacitance is dependent of the dielectric constant of the materials in and around the plates (in this case the plates are the etched, crossing lines on the backside of the glass). Also materials in the region of high electrical field (closer to the intersection) are more important. So all we have to do, to know how close a finger or a piece of wool (which both have a stronger reaction to electrical fields than air) is to our intersection, is to measure the capacitance between these lines and compare that to the capacitance we know from the case "just air in front of the glass".
How do we measure capacitance? In school I learned to measure capacitance by charging the capacitor with a known DC-voltage and a resistor and measure the time it takes to reach a certain voltage at the terminals. This method is obliviously not suited for this case because it would require precisely measuring lots of very small time intervals, since the capacitance of the crossing is quite small.
So we use a different approach: We simply apply a AC-voltage to the capacitors – 3V at 100kHz or so – and measure the current that is apparently flowing through the capacitor. The current measured is obviously not net displacement of charge, since there is no galvanic connection between the terminals of the small capacitor. Its just the moving there and fort of the elastically bound charges in the dielectric (imagine the single electrons moving half a micrometer in one direction and back 100000 times per second, that's the "current" we are talking about here).
This current increases with increasing dielectric constant of the surrounding materials. So when you move your finger toward one of these crossings the current apparently flowing through the crossing increases.
Something i forgot at first, is that this also works with conductors up to a certain point and not only dielectrics. This is because by replacing air with an conductor you are "closing the gap" and thus also increasing the capacitance between the plates a bit even though the conductor isn't even between the capacitive lines. You can test this with a spoon hold between two plastic objects: if the outward side of the spoon comes close enough to the touchscreen you also get a touch signal.
Now this still does not explain the bit about the grounding. Your finger does not actually have to be grounded to effect this rise in current. But when it is grounded the rise in current might be a bit higher than what would be expected since there is an additional connection formed for the current to take.
The electrons in your finger are not only influenced by the electrical field emitted by the matrix lines, but also by the field that is emitted by your body itself. When you touch the casing of the touchscreen the conductance of your body extends the field of the casing to your fingertip. Thus an additional way for the AC-current is formed through your body.
The importance of this additional current is strongly influenced by the design of the sensor. Some sensors may not react any different when you are in touch with its case than when you are not touching it. I tested the capacitive sensor of a Toshiba touchpad just now and for this one its one and the same weather I insulate the stumped carrot, which I used for the test, from my hand or not. So grounding is not important in this case.
However there is another effect in touchscreens and -pads that has to be accounted for: the number of matrix crossings affected by an object. Most capacitive touchsensors don't signal a touch if only one crossing shows an increased current or only do so if the current is a lot higher than normal.
This is a trick to increase fault tolerance of the sensor so it doesn't show every little piece of dirt or drop of moisture as a touch. Your finger always covers multiple crossings when you move it over the sensor thus it is a good idea to look for a patch of crossings with increased current, to find the exact location of you finger. This is usually done in software or a micro controller that interprets the raw inputs from the matrix. The strength of your touch is usually found by looking at the diameter of the patch with high current crossings. This works because your finger will flatten a bit when you press it against the surface.
A stylus may simulate the pressure signature by tuning a resonator inside it, which increases the current in a small spot a lot. But I believe there are different approaches to stylus input.
Please remember that there are a lot of capacitive touch sensors working with different systems. Some are drastically different than what I described here.
If I didn't explain something good enough please tell me.