Why does the LED need a resistor?

LEDs are designed to be driven at constant current. Simply connecting an LED to a 5v power supply is likely to result in too much current being drawn, potentially damagine either the power supply or the LED. A resistor is a very basic way of driving the LED safely, but for low power LEDs such as the ones we use here, it is generally sufficient.

Calculating the right resistance

To calculate the right resistor value, we simply use Ohm's law, $V=IR$. In this case, the voltage $V$ is 5v minus the operating voltage of your LED, typically 3.2v for a white LED, giving us 1.8v. The operating current of the LED I then sets the required resistance: $R=V/I$. For 30mA current, we therefore need $R=1.8/0.03=60$ Ohms. We specify 150 Ohms to drive the LED below its maximum current rating - we don't need the extra brightness, and it saves on power and heat.

Flicker and better drive electronics

Using a resistor only provides a constant current if the voltage is constant; it means any ripple in supply voltage will result in flickering of the LED. It is also difficult to perform wire-to-wire soldering reliably. For these reasons, future versions of the microscope will most likely use a small PCB to mount the LED and a constant current chip, which will more reliably provide a constant current. This assembly will then be easier to mount, and have a proper connector on it, thus eliminating the wire-to-wire soldering used currently.