Choosing the right capacitors and resistors for transimpedance circuit
I'm going to go ahead and combine my comments into an answer.


First is background knowledge.


I linked to this document in comments.  Here it is again.

You mention that you had already seen it.  I suggest you take a closer look.  It goes into the design process for selecting the circuit and parts.  It also has finished suggestions based on certain assumptions that come from knowledge about the subject.

You say you want to understand what it does and how it works - that document explains it, and goes into the science of what you are doing.


Parts


Let's take a look at just the transimpedance amplifier - that's the part you asked about.

Here's the transimpedance amplifier from the NXP document:



It explains how to calculate the expected output voltage from the diode current.  

The expected current can be found in the datasheet for the photodiode you are using.  

The datasheet gives you the expected current for a specific illumination intensity which will probably be higher than what you will get through a finger.  From that you can figure a resistor to start with.  Prototype with that value and see if you get enough voltage.  Expect to have to change it to get more voltage.

The NCP document  explains how to calculate the cutoff frequency for the low pass filter that is part of the transimpedance amplifier - and gives an example.  You will need to calculate the proper capacitor depending on the feedback resistor you selected for your expected current.

The selected cutoff wasn't chosen randomly.  NXP later mentions a bandpass filter that is implemented in software, and it has a similar cutoff on the high side.

It then details circuitry for cleaning up the signal.  It explains as simply as possible how to calculate the values used in the filters, and why each stage is used.


Calculations


NXP details how to calculate the oxygen saturation of the blood:



It sounds like you only need the pulse, in which case you don't need two different LED colors.  One is enough.

From each LED color, you get a signal like in that example.  The ratio of the two color signals gives you the oxygen saturation.

The signal from either signal contains the pulse.  Use the ATTiny comparator input and a timer to count time between pulses.  From time interval to pulse rate should be an easy calculation.



The most critical of the parts you asked about are C1 and R6.  Those are covered above.

The others are decoupling capacitors.  Anything from 100nF to a few uF will be fine.  

C2 in particular, though, should be placed right up close to the opamp power pins.  As physically close as possible, and with the shortest connections possible.  This is needed to help keep noise from the digital section (AATiny) from getting into the opamp.



So, take a good look.  Everything you need to know is in that document from NXP. 

If the equations don't make sense, ask about them here (a new question.)   Most of them are basic equations that you will need for general work with electronics.