Why these specific voltage divider resisters needed for Arduino Uno RX TX pins?
The output of a voltage divider is:

Vout = Vin * R2 / (R1 + R2)

Where R2 is the resistor in your case which is connected to ground.

See: Sparkfun - voltage dividers

Now in your example of 220 / 1k you have a multiplier of:

Vout = Vin * 1000 /1220 = Vin * 0.82


If both boards are running at 5V you don't need the voltage divider at all, but if you do use it the output would be:

Vout = 5 * 0.82 = 4.1


Now the Atmega328P chip on the Uno has the electrical characteristics defined in the datasheet, amongst others is that a digital pin high has to have a minimum of 0.6 * Vcc (if Vcc is in the range 2.4V to 5.5V)

So if Vcc is 5V (which it is on the Uno) you have to have 0.6 * 5 (that is, 3V) as the minimum for it to be recognised as high.

So, using the voltage divider of 220/1k you have a high still being 4.1V which is in the range 3 to 5V.

However if you have the board set to 3.3V then the output of the voltage divider will be 3.3 * 0.82 = 2.76V which is too low to be recognized as high.

For sending serial from a 3.3V board to a 5V board, simply omit the voltage divider, and then the 3.3V (for high) is in range (it has to be 3+ volts).

However for sending from a 5V board to a 3.3V board you do need a voltage divider (or level shifter) because 5V is too much for a 3.3V board. You can use a voltage divider calculator to simplify the maths.

Something like 1k/1.8k would be about right.

Vout = 5 * 1800 / 2800 = 3.21




According to the documentation for the Parallax WX ESP8266 Wi-Fi Module:


  (9) Level shifters and buffers
  
  To ensure maximum compatibility with a wide range of 3.3V and 5V microcontrollers and development boards, the Parallax Wi-Fi module includes voltage level shifters and buffers to ensure that the logic voltage levels at the SIP header are set by the VIN voltage level. If you
  supply 3.3V to VIN, then the IO pins logic voltage level will be 3.3V. If you supply 5V to VIN, then the IO pins logic voltage level will be 5V


I don't see why you need to use a voltage divider or level shifter in the first place.



References


Atmega328P datasheet (32.9 MB)
Sparkfun - voltage dividers
Voltage divider calculator
Another voltage divider calculator
Voltage dividers - Wikipedia