Features :
For Arduino UNO R3 and the compatible board can directly connect to this module.
Interface logic voltage: 3.3V / 5V compatible(On-board level shift circuit)
Working voltage: 4.5V - 5.5V (On-board 3.3v LDO Regulator)
I think you need this connection but I have never used the board in your picture.
5V UCC
GND GND
TX RX
RX TX
The switch must be in the UART position (not program). The baud rate should be 115200. If still not working, try different baud rates starting from 9600. Send AT followed by one of the following line ending options: CR, LF, CR-LF.
What is Arduino Board Manager? (ESP8266 or Arduino UNO)
The board manager configures the IDE to work with different Arduino boards. Some boards such as the ESP8266 do not have built-in support but support can be added. See the ESP8266 Arduino site for installation details.
WARNING: Loading new firmware into the ESP8266 will erase the AT command firmware. I suggest getting an general purpose ESP12 board such as a Wemos D1 mini or NodeMCU if you wish to use the Arduino IDE with an ESP8266.