A DC power supply with the right voltage is okay, also that one. You did connect it to the barrel jack ?
The most likely reason is that your Arduino board draws too much current for the voltage regulator on the Arduino board. Suppose you have an Arduino Uno with a servo motor or a lot of leds and they require 500mA. The voltage regulator would heat up with (9V-5V)*500mA = 2 Watt.
Can you find the voltage regulator on the Arduino board ? You can check if it is getting to hot by holding your finger on the voltage regulator. If you can keep your finger on it, then it is okay.
The voltage regulator should reduce the current too prevent that it gets too hot, but it is not hard to blow the voltage regulator despite that.
The DC power supply could be bad quality and could have an output voltage that is too high or has voltage peaks, or there is a leakage current from the mains. Do you have another power supply ?
I use a good quality and certified DC power supply of 7.5V. In the photo I see all the right certification labels, but I don't know if they are real.
What was connected to your Arduino board ? If nothing was connected, then you are in trouble, and it could be faulty DC power supply or a bad Arduino board.