First I used wires from arduino starter kit, I thought they are not good enough so i used different ones, for example from ethernet cable but it didnt help.
Then I tried to change power supply from USB (as u can see in the video) to battery charger with 2.1mm power plug (i connected it to arduino) and after that i connected EXT_PWR from the shield to a lab power supply. It also didnt help.
I think the engines are fine, when i connect them directly to 3.3V and Gnd in arduino they are working really good.
I think there is something wrong with the shield. When i wrote a code to control only one engine, sometimes the other one was also rotating.
The shield uses the L293D or the newer SN754410. The minimum motor supply voltage is 4.5V. More realistic voltages are 6-7 volts depending on the motor. The motor will not receive the full voltage.
I cannot tell from the video where you are getting the motor power. A 'battery charger' voltage may droop too low under significant load. The Arduino itself may only draw 50-100ma but the motors may surge on startup to 500ma or more. The surge would cause a 'brown out' resetting the Arduino, perhaps repeatedly.
If you have connected the power jumper, then pull it off. Connect a reasonable power source to the External Power block and try again. A battery pack such as 4 alkalines or a 2 cell LiPo is a good choice.
If it works OK, then you can reinstall the jumper and use the batteries to supply both Arduino and motors.
I am making assumptions since all I can see is a somewhat fuzzy video.
Sorry to disappoint you.