There are only 3 connections: GND, OUT, +Vin
So the first question is what is the safe supply voltage range to connect to the +Vin pin. According to the first Google result, the minimum is 5 V:
So to power it you need to connect the GND pin from your Arduino to the GND pin on the HW-MS03 and the 5V pin from your Arduino to the +Vin pin on the HW-MS03.
So the way this is going to work is like the PIR sensors. When the module doesn't detect movement, the OUT pin will be in one state (LOW or HIGH, I'm not sure) when it detects movement, the OUT pin will switch to the other state for 5 seconds (according to the same video: #135 Radar Sensors / Switches: Comparison and Tests - YouTube), before switching back (unless movement has continued to be detected).
Since it runs at 5 V, likely the output is also 5 V. Since your ESP8266 is a 3.3 V device, you will probably want to convert the 5 V output of the module to 3.3 V. You can do this with a resistive voltage divider. So connect the OUT pin on the HW-MS03 to any free IO pin on your Arduino via the voltage divider.
You really don't need a library to use with this sensor because it's so simple. Likely there is some PIR sensor library for Arduino that will work just as well with this sensor, but anyone who would write such a library is probably a beginner so there is no guarantee that the code will be any good. The most basic code you could use to test the sensor looks like this:
const byte motionSensorPin = 3;
void setup() {
 Serial.begin(9600);
}
void loop() {
 Serial.println(digitalRead(motionSensorPin));
}
This is configured for the motion sensor's OUT pin to be connected to pin 3 on your ESP8266, but you can adapt it to any pin you like. Note that on some ESP8266 boards like the WeMos D1 Mini, you need to use the pin names like D3 in order for them to match to the labels on the silkscreen of the board.
Upload that sketch to your ESP8266, open Serial Monitor, then make sure the baud rate is set to 9600. you will see a string of 0 or 1 scrolling down the screen depending on whether the sensor has detected motion or not. You should be able to go from there to write code for your particular application.