Learn Hardcore German: Die Laterne am Tor flackert, weil die Batterie fast leer ist. - The lantern at the gate flickers because the battery is almost empty.

an is a two-way preposition that can govern accusative (direction) or dative (location).
Here it indicates location (“at the gate”), so you use dative dem Tor, which contracts to am Tor.

Why does an take the dative case in this sentence?

German two-way prepositions (an, in, auf, etc.) use:

accusative when showing movement toward something,
dative when showing a static location.
Since the lantern is simply located “at the gate,” there’s no movement, so you use the dative case.

Why is there a comma before weil?

weil introduces a subordinate (dependent) clause in German. The rule is:

You always separate a main clause and a subordinate clause with a comma.

Why does the verb ist appear at the end of the clause introduced by weil?

In German subordinate clauses (introduced by conjunctions like weil), the finite verb moves to the very end of that clause. So you get:
…, weil die Batterie fast leer ist.

What does fast mean here? It looks like the English word “fast.”

In this context fast means almost, not “quickly.” You could also say beinahe leer (“nearly empty”).

Why is die used for both Laterne and Batterie?

Both Laterne and Batterie are feminine nouns in German. A quick hint: many nouns ending in -e or -ie are feminine.

Can I use da instead of weil in this sentence?

Yes, you can say Die Laterne am Tor flackert, da die Batterie fast leer ist.

da also means “because,” but it’s slightly more formal or background-giving than the more direct weil.

Can I start with the weil clause for emphasis, as in Weil die Batterie fast leer ist, flackert die Laterne am Tor?

Absolutely. If you front the subordinate clause, you write:
Weil die Batterie fast leer ist, flackert die Laterne am Tor.
After the comma, the main clause follows. If its Vorfeld (first slot) is empty, the verb appears first: flackert.