Flash your ESP8266 with Micropython. Follow instructions from here:

Edit the main.py file to connect to your own Wifi

In the main.py define the GPIO pins you connect the devices to.
As it is when you download it:
- DHT22 connected to GPIO16
- SSD1306 connected using I2C: SDA to GPIO4, SCL to GPIO5
- Output to the led of the ESP8266 (GPIO2)

In the server side: install a MQTT Broker (I've used Mosquitto on Linux)

In the main.py file, edit the address of the MQTT broker (server) you will be sending the info of Temperature and Humidity to.

Transfer the files from the project to the ESP8266. My advise is to use the Webrepl file transfer option.
Follow instructions from here: http://docs.micropython.org/en/latest/esp8266/esp8266/tutorial/repl.html

Edit your boot.py in the ESP8266 so it won't autostart the webrepl service and thus you have more memory available.

In the server side: install and configure OpenHab Runtime with MQTT binding.
Follow instructions from: https://github.com/openhab/openhab/wiki/MQTT-Binding