Do I need the Relay-Board?

TL;DR: Yes, you do!

The idea behind the 4 channel relay board is to allow the SMuFF firmware to switch between the 3D printers and the SMuFF Feeder stepper driver on demand and thus realize a "Shared Stepper".
This has the big advantage that the SMuFF gains far more control over the loading / unloading process and enables it to handle issues on its own if necessary, without the need for a complex communication protocol between SMuFF and 3D Printer controller - which of course would require a modification of the firmware on your 3D printer. 

Because many 3D printer controllers come with integrated stepper drivers that can't be easily removed, the phase signal outputs of the stepper driver (1A/2A, 1B/2B) need to be switched. Those signals run higher voltages and higher currents than the usual TTL ICs (usually +12/+24V, 1-2A) and must be separated in order not to interfere with each other, a relay is needed to switch between them.

If you're not familiar what a relay is and how it works, here's a video explaining it all:

Keep in mind: There are two potential downsides when using a servo in this scenario:

Please keep in mind: In contrast to other (formerly optional) components, the relay board, although it still can be configured via the "Shared Stepper" option, is not as much optional anymore. In future releases of the SMuFF firmware it'll become mandatory, hence my recommendation is you better get one right from the start.

More details on this topic

Since a relay is an electrically controlled switch alternating between two poles, we need a 4 channel relay (or relay board) to switch all 4 wires of a NEMA stepper motor at the same time.

There are a plenty of different modules available in the maker space, built to be controlled by a controller such as the Arduino. Here are two of such 4 channel relay modules, which both can be used for this purpose on the SMuFF.

In case you've seen my Mini-Relay-Board PCB or the SMuFF Backbone board, you may have noticed only two relays. This is because this type of relay combines two of those switches in one package, which saves on space for the PCB but since those are also physically smaller, they have a lower switching current rating. One can also get 4 relays in one single package, which are not as common and thus significantly more expensive than the whole relay board.

One thing all the boards have in common is: Each of the relays is a SPDT, which stands for Single Pole Dual Throw.

This means that the input (voltage) is being switched between two outputs. For easier distinction those are commonly named COM for the input (voltage), N.C. (Normally Connected) pole, which is active as long as the module is either not being powered or its IN signal is in the "OFF" state and the N.O. (Normally Open) pole, which isn't active when the board is not being powered but becomes active when its IN Signal is in "ON" state.

Since current flows in either direction, the COM can also become an output, which is what we're using to connect the stepper motor to and the state of the IN signal determines whether the 3D printer or the SMuFF has access to it.

This is exactly what you see in my Assembly Manual:

On those standard off-the-shelf relay modules, each of the 4 wires on the SMuFF Feeder stepper motor is connected to the COM terminal. The 4 wires of the 3D printers E-Stepper cable are going to the N.C. terminals, which means: In case the SMuFF (i.e. the relay board) is not powered, your 3D printer has full control over the Feeder. All 4 wires coming from the SMuFF Feeder stepper driver are going to the N.O. terminals. Whether this is the Z or Y-Axis driver header depends on the use case (a.k.a DDE or Bowden mode).

Please notice: One really important fact to keep an eye on is that you don't mix up the coil wires (A/B) of your stepper motor. Otherwise it'll cause the stepper motor to rattle rather then to spin.
Unfortunately, there's no standard  which pin is supposed to be which coil, and hence you might end up with different layouts on the controller boards, as well as the stepper motor itself, e.g.: 

A 1:1 connection of the wires shown above will definitely cause such effect.

To keep the overview, I recommend using the same color scheme on all stepper cables, as you see in the picture above. Each wire on the screw terminals of one particular relay should have the same color. This avoids mixing them up. If you put ferrules on those wires (which is always a good practice for safety reasons) using different colors, you can easily tell at first glance which wire belongs to which terminal. In the picture above you see all the black ferrules go into the middle position (i.e. COM), all red ones are on the left (N.C.) and all the blue ones on the right (N.O.).

While controller boards are mostly well documented regarding the pinout, some stepper motors aren't and you'll need to find out the coil pairs (A/B) yourself.
The easiest way to achieve this, is measuring the continuity/resistance of the pin pairs on the stepper motor 6-pin header using a voltmeter. Those two that show 0 Ohm are belong to the same coil.

On the other side (opposite to the relay terminals) are the MCU control inputs, which are used to switch the relay. Since those relays are usually supposed to control different devices/circuits, each of them has it's own input (IN1-IN4) but one VCC and GND input to power the board / relays. The VCC voltage depends on the relay specification, which is 5V in case of those standard relay boards meant to be connected to an Arduino MCU (or similar).

As we need all relays to switch concurrently, we only need one signal on the SMuFF controller to handle them. Hence, we can short IN1-IN4, which is easiest by soldering a bridging wire at the bottom of the relay PCB.