How to determine necessary electrical components for breakout board
Many times, what you are calling a breakout board is in fact a complete circuit.  

The other parts are all the things needed to make it work - often implemented from a recommended circuit in the datasheet.  

A breakout board in the usual sense is just a board where you can attach a small or complicated chip and have manageable connections using wires or breadboard pins.



You figure out what goes on the board by looking at what it takes to make a functional circuit (supply voltage, needed control signals, additional components, etc.) you get that from the datasheet.  

Then, you look and see what your main board (Arduino or whatever) can supply.  

Suppose, for example, your external chip needs 3.3V.  Well, there's 3.3V available from the Arduino.  Fine.  Only, the chip is a 24 bit ADC.  The 3.3V from the Arduino is really nasty, so either include a separate regulator on the breakout board, or heavily filter the 3.3V from the Arduino.

Now, the 3.3V power for the ADC means that it also uses 3.3V logic levels.  Can the Arduino accept those? Can the ADC tolerate the 5V logic level from the Arduino? If not, include logic level shifting on your breakout.

Then, you get general design principles.

In our example of an ADC, we will need decoupling capacitors on the ADC power pins.  Pretty much any chip needs this.  Either to prevent its switching noise from disturbing other parts, or to prevent noise from other parts causing a problem to the chip you are looking at.

So, there's a lot that goes into it. There's no simple "sprinkle a bunch of resistors and capacitors over it like pixie dust" solution.

What you need for an ADC is different from what you need for a relay driver is different than what you need for a sensor is different from other stuff.

Case by case, whatever is needed/required to make it work and make it safe.



In response to comments from OP:

Many datasheets include example circuits. Often times, the examples are also the circuits used when the manufacturer measured the real performance of the chip - following those recommendations should get performance as in the data tables in the datasheet. Good datasheets also mention why certain parts were chosen.

So, when designing a breakout board (or any other circuit using a particular IC,) you start by looking the datasheet and the examples it gives.

Regardless of whether or not the datasheet provides an example, you will have match the connections of the breakout to what the base device expects and provides.  

Since already have your base device selected (Arduino or Pi or whatever,) get familiar with it first - what signal levels it expects, what power it can provide to external devices, how it can talk to external circuits, etc.  Then, you look for an IC or circuit that provides whatever function you need.  Find one that matches your functional requirements, and that also can easily talk to your base device.

Maybe you will find that no IC that meets the functional requirements can easily talk to your base device.  Well, you can either add circuitry to make it possible (say, level shifting so a 1.8V IC can talk to a 5V Arduino.) Or, you back up and reconsider the base device - it might be simpler all around to use something that can natively address your external device.

So, pick one end and work it through from there.  You might end up making a couple of loops back and forth - well, it happens.

You can use programs like LTSpice to simulate circuits before you build them, but they won't all contain models of all ICs.  I'm not a big fan of them.  They all have inaccuracies somewhere, and as a beginner (or hobbyist like me) you won'necessarily know when a circuit is failing because it is bad or because you've hit one of those odd wrinkles in the software.

Pick something interesting, and see what it would take to accomplish it.  Use available modules to begin with - this gets you some success and visible progress.  

See how the modules function, and how they are designed.

On your next project, consider something that needs a simple external circuit.  Look at how others accomplish it, then put together a circuit that you think will work.  Build it on a breadboard or perfboard to test it.  Fix it, improve it, have a PCB made so that you can install it permanently in your device.

You can post circuits here and ask for help when they don't work (or don't work right.) But, you should post your circuit diagrams (and often times the layout or a picture of your breadboard) when you do.  Ask direct questions rather than open ended ones.  (Bad: "Critique this five page circuit diagram." Good: "This amplifier is oscillating.  Here's the circuit.  What have I done that makes it unstable?")