The Matter Board is a device that moves the plot in PSO2. You will need to complete objectives on it in order to continue the story along.
You are eligible to receive your first Matter Board as soon as you finish the tutorial mission. You’ll need to head to the shopping floor of the Arks Ship. Use an elevator to get there.
When youre there, jump down to the ground floor. The person you will need to look for is someone called Shion. I’ve circled her in red in the below picture.
That’s all! You now have a Matter Board.
When the cutscene ends, you’ll have to pick a node to activate. There’s only one choice at this point so just hit enter and click the left button of the window that pops up.