What was the process of calculating how much darkmatter/matter/dark energy the observable universe consists of and how did it come at this conclusion In 2015 the Planck satellite consortium released an analysis of data collected by the Planck satellite that showed this percentage break down of the universe’s mass/energy.

Ordinary matter: 4.9%

Dark Matter: 26.8%,

Dark Energy: 68.3%

How did they calculate the "stuff" of the observable universe and how did they come to this conclusion?

I assume  this took some tough calculations and some work with a supercomputer but i wonder what the entire process was.
We are not actually measuring the $\Omega_r$,$\Omega_m$,$\Omega_{\Lambda}$.
At first, we measure the distance modulus to certain celestial objects by using the standart candles in the astronomy.
This distance can be calculated from,
$$d_L=\frac {c} {H_0}z(1+\frac {1-q_0} {2}z)\,\,\, (Eqn.1)$$
Here $q_0$ represents the deceleration parameter. Now I am not going to go in mathematical details but you should now that, using the acceleration equation we can write $q_0$ in terms of $\Omega_r$,$\Omega_m$,$\Omega_{\Lambda}$.
$$q_0=1/2 \sum \Omega_{w,0}(1+3w)$$  and for matter ,$w=0$, for radiation $w=1/3$ and for dark energt $w=-1$ we get,
$$q_0=\Omega_r+1/2\Omega_m - \Omega_{\Lambda}\,\,\, (Eqn.2)$$
And the relationship between distance modulus $(m-M)$ and $d_L$ can be written as,
$$m-M=5log_{10}(d_L/1Mpc)+25\,\,\, (Eqn.3)$$
If we combine Eqn.1 and Eqn.3 we get a relationship between distance modulus and z.
We plot the $m-M$ for distant objects with respect to $z$. From the slope of the graph we can determine the $H_0$ (for $z\rightarrow 0$) And for large z,  deviation of the plot from the straight line gives us to determine the $q_0$.
Now we have experimental data. Then we put different values for $q_0$ or equally $\Omega_m$ and $\Omega_{\Lambda}$ (with respect to Eqn.2) to determine the best fit for the graph (since we can take $\Omega_r=0$)


Edit: I clearly missed the &quot;The Planck data&quot; in the OP's question but its the same general idea. For different density parameters, the CMBR would look different, like the graphs that I shared are different for different density parameters. So by looking at the CMBR, we can determine the density parameters and which one fits well to the CMBR.

Further Note: I want to also suggest that the $\Omega_b$ can be calculated from the early universe by measuring the amount of deuterium and other elements present in primordial gas clouds.
And observering the rotational curves on the galaxies we can have approximate values on the $\Omega_{dm}$. (finding the dark matter density)
We cannot deduce the $\Omega_m$ and $\Omega_{\Lambda}$ just by looking at the Supernova data, for example the supernova data is also consistent with $\Omega_m=0$ and $\Omega_{\Lambda}=0.4$ or with $\Omega_m=1$ and $\Omega_{\Lambda}=1.7$