You have connected the tx of the Arduino to the tx of the ESP8266 and the rx of the Arduino to the rx of the ESP8266.
What is happening when you type AT in the Serial Monitor is that you are communicating directly with the ESP8266, and not with the Arduino.
You need to connect the rx of the Arduino to the tx of the ESP8266 and the tx of the Arduino to the rx of the ESP8266.
This has asked and answered a number of times, such as:
Arduino Uno and ESP8266 serial monitor issue
Arduino & ESP8266
Arduino Nano v3.0 with ESP8266
ESP 8266 does not respond at all
Your ESP8266 definitely uses a baud rate of 115200 (as confirmed by your first screen shot), so you don't need to change that when experimenting.