The Avatar is tasked with bringing balance to the world, and that often takes him across a wide array of political, cultural, and spiritual challenges. As the last Airbender, Aang carries an incredible amount of responsibility with his role, and the arrow tattoos are one of the key visual cues to his status as the Avatar that help him to navigate these complex issues throughout the series.
The arrows also represent Aang's connection to the spirit world. Throughout the series, Aang encounters spirits (most notably the lionturtles) and has conversations with them. They have a profound influence on him and his philosophy, and he believes that there is an underlying energy that connects all living things.
Similarly, Aang carries on the culture and philosophy of the Air Nomads, who believed that all life was sacred. This belief is a huge part of his outlook on the world, and it informs his philosophy on bending and fighting. It is also the reason that he refuses to use violence unless absolutely necessary.
As the Avatar, Aang is connected to a long line of previous Avatars who have mastered all four elements and worked to keep balance in the world. They are a source of wisdom and guidance for Aang, and the glowing arrows in his stance are a reminder of this connection. This is a subtle but effective way to underscore the importance of tradition and the interconnectedness of all things.