Use of connecting capacitor in I to V converter
The basic transimpedance amplifier is, as you've properly stated, just a simple resistor feedback.



The opamp, in a simplistic model, will source the voltage on the output necessary to drop your I_in across R. For DC or near-DC signals you only care about the resistance R when setting your gain factor. AC signals being sent across the transimpedance amp will see  a slightly different picture.

Internally, the opamp has some capacitance between the input and ouput nodes. This establishes the maximum bandwidth of the op-amp. Doing nothing (jsut leaving the resistor) you will see that frequencies above the Gain Bandwidth Product of your opamp (17mhz for yours) will be shunted. However, often times its useful to shape the bandwidth of you amplifier in order to filter out signals that are undesirable, placing a capacitor across the feedback reduces the impedance seen by high frequency signals and thus reduces its ouput gain relative to the dc level signal (which only sees the resistor as a feedback current path). Thus you have a basic LP filter built into your transimpedance amplifier. Additionally, as was pointed out in the comments, many common sources used in conjunction with transimpedance amps (like photodiodes) have their own internal capacitance which act as their own filter and can change the frequency response even in the operating range, by placing a capacitor across the feedback you can have a more predictable frequency response and essentially take the capacitance of the source out of the picture.

The power supply capacitors are known as "decoupling capacitors" they serve several purposes.


They filter the power supply from high frequency transients and coupled noise by providing them a low impedance path to ground
They act as a electron "bucket", When the opamp swings from one end to another it starts drawing current from the power-supply lines, the power supply doesn't respond immediately, and stray inductances on the power supply line can prevent the opamp from drawing the current it needs and the voltage will drop. Like a backup battery the capacitance will provide the op-amp with its instantaneous power requirements as its current demands increase, as such they are usually placed as close as possible to opamp when laying out the circuit


Often times you will see two capacitors in parallel to ground. One high valued capacitor to act as the "backup" and another low-valued capacitor as the HF shunt to ground. WHile this may seem pointless (if capacitors add in parallel why not just use a larger one?) it has its benefits, while the capacitance of the capacitors adds in parallel, their small (but finite) resistance (from the wire leads) will be reduced and more importantly, as was pointed out in the comments, the inductance will also be reduced. So the impedance to ground for HF signals is much lower with two individual capacitors vs one larger one. If no HF supply filtering is needed, a larger capacitor alone is enough,