the Espruino has a linear Voltage regulator (MIC5205) so since the electronics need 3.3V as power supply, the excess is converted to heat in the voltage regulator. Assuming your whole setup takes about 200mA current, for a 9V battery this means 5.7V * 0.2A = 1140mW = 1.14W of heat is being generated by the voltage regulator (9V - 3.3V = 5.7 V).
Since it's mounted on the PCB without really a heat sink, I am guessing it gets really hot and goes into thermal shutdown (The MIC5205 has an internal thermal protection, see the datasheet: http://www.micrel.com/_PDF/mic5205.pdf There are also some more detailed thermal calculations on pages 8/9 of the datasheet).
At least that is my guess as to what is happening. The other scenario is that your battery is already too weak to supply the current for the Espruino + BT + CC300 module, but you could easily measure that by checking the battery voltage when you have everything connected.
I would probably use a rechargeable battery (LiPo) or 3 NiMh cells to power the Espruino. A 9V battery has inherently a pretty low energy density. You could also use a high-efficiency switching regulator circuit to power the Espruino if you really have to use 9V batteries.
Hope that helps,