OK let's first understand units. The decibel (dB) is a base-10 log scale without units and dBm is a similar decibel scale for power referenced to 1 milliwatt. They also include a factor of 10, so for example 10 dB is a ratio of 10^1, 20 dB is a ratio of 10^2, etc, while 10 and 20 dBm would be 10 mW and 100 mW.
But in the block quote, they use dBW instead of mW, so $log_{10}(22)$ = 1.342 and it's shown as 13.42 dbW. While dBm is more common, let's stick with Watts here.
The standard way to calculate the received power on Earth is to use a link budget calculation. This is one way to calculate the received power in a standardized format so Engineers can understand each part of the link separately, and to share the information with each other. Since the calculation is a series of multiplications and division, when you use dB, these become addition and subtraction of logarithms. I'm going to leave out the smaller corrections from the big equation shown here since this is an approximate calculation.
$$ P_{RX} = P_{TX} + G_{TX} - L_{FS} + G_{RX} $$
- $P_{RX}$: received power on Earth
- $P_{TX}$: transmitted power by Voyager
- $G_{TX}$: Gain of Voyagers transmitting antenna (compared to isotropic)
- $L_{FS}$: Free space Loss, what we usually call $1/r^2$
- $G_{RX}$: Gain of Earth's receiving antenna (compared to isotropic)
We know that $P_{TX}$ is 13.4 dBW already, and on page 17 of the DESCANSO Design and Performance Summary Series Article 4: Voyager Telecommunications we can see that Voyager's high gain (X-band, around 8.4 GHz) antenna has a $G_{TX}$ gain of 48 dBi, where the "i" means relative to a theoretical isotropic radiator.
The gain of the receiving dish antenna $G_{RX}$ can be caluladed (from here) as
$$G_{Dish} \sim \left( \frac{\pi d}{\lambda} \right)^2 e_A$$
where $d$ is the diameter of the dish, $\lambda$ is the wavelength, which is the speed of light of 3E+08 m/s divided by the frequency of 8.4E+09 Hz or about 0.036 meter (3.6 centimeters), and $e_A$ is some aperture efficiency term between 0 and 1 for a realistic dish, which we'll set to 1 to make things simple. For the Deep Space Network's largest diameter dish antenna of 70 meters, this becomes about 1.9E+07 which after applying $10 \times \log_{10}$ becomes about 73 dB.
The Free Space path loss is calculated by calculating the fraction of an expanding spherical wave (from an isotropic radiator) that would be received by an area similar to one square wavelength. The exact equation in dB is:
$$L_{FS} = 20 \times \log_{10}\left( 4 \pi \frac{R}{\lambda} \right).$$
The reason the fraction flipped, but a minus sign did not appear outside is because by convention, loss is expressed in positive dB, and then subtracted by the minus sign in the "master equation". Currently Voyager 1 is about 2.1E+13 meters (yes, 21 billion kilometers!) away, so $L_{FS}$ is about 7.3E+16 or 317 dB.
$$ P_{RX} \ dBW = 13.4 \ dBW + 48 \ dB - 317 \ dB + 73 \ dB = -182.6 \ dBW$$
which is darn close to the -181.4 dBW shown in the question!
When receiving the signal, the limit to the data rate is the ratio of received signal power to the total noise power (received plus system). We calculate both for a fixed range in frequency, which should be roughly the bandwidth that Voyager is using.
For an effective receiver temperature of say 20 Kelvin, the noise equivalent power will be about $k_B T \times \Delta f$ where $k_B$ is the Boltzmann constant.
I'll do some handwaving${}^†$ here and just estimate the bandwidth used by Voyager's spread-spectrum transmission to be about 1 kHz, a few times larger than the quoted bit rate of 160 bits/second would require. That makes the noise effective power about 1.3E-20 Watts or -199 dBW, and that gives a signal to noise ratio (S/N) of -182.6 dBW minus -199 dBW of 16.4 dB, which is more than ample for good reception!
update: Thanks to @TomSpilker's careful review: That makes the noise effective power about 2.7E-19 or -182.6 dBW minus -185.6 dBW = 3 dB, which is sufficient when used with some combination of redundancy and error correction.
†edit: @Hobbes' comment points out that I don't really know if Voyager uses spread-spectrum for data communications or not, since I've recently asked Have deep-space spacecraft always used some form of spread-spectrum for data downlink?. I had assumed that it would have been used to improve the S/N ratio, but that was a groundless assumption. Stay tuned for more updates!
