What information that I did find for the LGM12864B touchscreen does not seem to match the one you have because the datasheets explanation for how the screen works is through a protocol and doesn't use a resistive response.
So I'll answer for how the resistive touchscreen works in your case. When you press on a resistive touchscreen, the pressure will introduce a dynamic voltage divider between the two X nodes, and the two Y nodes. This image may be useful for understanding:
You can see on page 3 of the document that Gurn64 provided that you can read the touchscreen in 2 steps:
First, to determine the horizontal position, set Y+ to be 5V and Y- to be ground. X- is essentially floating, and X+ is now your output. Measure X+ and the ratio of X+ to 5V is the ratio of the distance from the bottom of the screen to the length of the height of the screen.
The vertical sensing is similar. Y+ and Y- are no longer 5V and ground respectively. X+ and X- are 5V and ground respectively, and Y+ becomes your output like X+ was before. The explanation is the above picture is pretty clear.
Also, another note is to remove the 100Ohm resistors that you currently have hooked up, they don't aid in the procedure of determining touch positions.