How does Open Assistant work?
I’m not sure how the question is meant. I see two ways how your questions could be interpreted and answered: 

If the question is meant how to make the software work to give your assistance: As of today (Febuary 2023), Open Assistant doesn’t work in the sense that it offers any assistance to the user. That is because it is still under development.
On the project's GitHub page [https://github.com/LAION-AI/Open-Assistant] it is written that the vision for open assistant is the following: 
We want to build the assistant of the future, able to not only write email and cover letters, but do meaningful work, use APIs, dynamically research information, and much more, with the ability to be personalized and extended by anyone. And we want to do this in a way that is open and accessible, which means we must not only build a great assistant, but also make it small and efficient enough to run on consumer hardware.

If the question was rather meant to question how Open Assistant works on a technical level: Open Assistant is based on a large language model that is enhanced and re-trained to work chat based, meaning that it offers language interaction as you would be accustomed to when talking to another person. The language model itself is currently implemented in form a specific artificial intelligence system called a neural network trained on a large corpus of texts. The training for chat purposes is an ongoing endeavor involving human feedback to teach the artificial intelligence to give high quality replies on a large variety of questions and using the feedback to use it’s learned data and concepts to give useful replies as humans would expect.
In the future, Open Assistant is also supposed to be able to dynamically retrieve information to answer the questions even better which is in constast to other current chat-based assistants like ChatGPT from OpenAI [https://openai.com/blog/chatgpt/]. 
The source code is open source and documented and available on the GitHub page of the project: [https://github.com/LAION-AI/Open-Assistant]