How to connect HC-06 BT module without pins?
The small green PCB is the real HC-06 board. The blue board has 2 functions:


It breaks out the contact pads of the HC-06 to the bigger pins.
It has a voltage regulator on it so that you can provide it with 5V power, which then get's converted to 3.3V for the HC-06 (which is a 3.3V device).


The next steps depend on what Arduino board you have:


If you have a 3.3V Arduino Board, you can directly connect the pads on the HC-06 (by soldering pins or wires to them) to the corresponding Arduino pins:

Arduino &lt;--&gt; HC-06
  3.3V  &lt;--&gt; 3.3V
  GND   &lt;--&gt; GND
  TX    &lt;--&gt; RX
  RX    &lt;--&gt; TX


Also you can connect a fitting LED to the LED pin. I spontaneously cannot find data about the max current of this pin. But I guess it is in the range of 10 to 20 mA. Use a corresponding combination of an LED and a resistor.
If you have a 5V Arduino, you need to consider the voltage difference. On the Uno and Nano you have a 3.3V regulator on board. The regulator itself is rated for 150mA, but since it is not cooled, the limit is more like 50mA (see this forum thread). This datasheet of the HC-06 states, that it draws at max 30-40mA, so you can get away without an extra 3.3V regulator, just by connecting the both 3.3V pins on Arduino and HC-06. If the voltage regulator overheats despite being under 50mA, you need to buy an extra 3.3V voltage regulator, that can handle the current.

Don't forget to use a voltage divider on the Arduino-TX --> HC-06 RX line to divide the Arduinos 5V down to 3.3V. This line isn't even regulated by the blue breakout board, so you would need this voltage divider either way.

Ground connection and the LED are the same as in the first case.


So all in all you can still use the HC-06, just with a little bit less ease than with the blue breakout board.