Learn Italian: Come si avanza su queste caselle? - How does one advance on these squares?

It acts as the impersonal si. Instead of asking how a specific person moves, it asks how anyone moves. In English, we usually translate this as "How do you advance?" (meaning players in general) or "How does one advance?".

What exactly is a casella?

Casella (plural: caselle) refers to a small box, slot, or square on a grid. In the context of moving or advancing, it means a space or square on a board game.

Why is it queste caselle and not questi caselle?

Because caselle is a feminine plural noun (from the singular casella). Therefore, the word for "these" must also take its feminine plural form, which is queste.

How would I use this lesson's grammar to ask "On which squares does one advance?"

You would use quali, the plural form of quale (which), so that it agrees with the plural noun caselle. The sentence would become: Su quali caselle si avanza?