This looks like a good first project!
It won't be as simple as just hooking up each of the servo pins to the module. The servo pins output a pwm signal, but the lcd display expects simple digital signals.
The first thing that you'll want to do on any hardware project is locate the relevant datasheet. Here is the datasheet for this screen.
I believe Arduino has a built-in Liquid Crystal reference and library that can be used to write to all kinds of 16-pin LCD screens. You won't be able to use their exact code because it's in C but it will at least give you a starting point for understanding how to make the screen work from code once the hardware is connected properly.
In terms of pin connections, we will be connecting up power, ground and six GPIOs. One common mistake is to use the wrong logic level between microcontrollers and peripherals. A logic level is simply the voltage of a pin when it's in the "high" state. Tessel is a 3.3V device but this screen may not be. If we take a peek at the datasheet, it looks like the screen expects to be powered with 5V (page 2, "VDD") so if Tessel is powered via USB, we can hook up the VIN pin on the GPIO bank to the power pin on the LCD screen (which your schematic indicates is the 2nd and penultimate pin).
Lucky for us, this screen also accepts an "input high voltage" (high digital signal) of 2.2 to VDD so we can use any of the GPIO pins on Tessel which run at 3.3V for input-only pins. Unfortunately, we'll have to convert the voltage of pins that may send data to the Tessel from the screen down to 3.3V because the datasheet indicates that they will be between 2.4V and 5V. If they are greater than 3.3V, then they will damage Tessel. You will want to connect all of the data pins to something like this converter.
We can find the descriptions of what each of the signal pins do in the datasheet as well:
"Register Select" basically dictates whether you are sending an instruction (eg. "write this to the screen") or data to be written on the screen. The rest of the registers are fairly self explanatory and the datasheet goes into more detail into the instructions (page 7 and below).
I hope that helps! Let me know if there is anything you're still confused about.