Den Restmüll bringe ich nur einmal pro Woche raus, weil die Tonne sonst zu schnell voll wird.

German often puts whatever you want to emphasize or connect to the previous context in position 1 (the “topic” position). Here, Den Restmüll is fronted for emphasis/contrast (e.g., “As for residual waste…”).
Even though something other than the subject is first, the verb still stays in position 2 in a main clause:

• Den Restmüll (pos. 1) + bringe (pos. 2) + ich (later)

Because bringen is a verb that takes a direct object (what you bring), and direct objects are usually accusative.

• der Restmüll (nom.) → den Restmüll (acc.)

You could ask: Wen oder was bringe ich raus? → Den Restmüll.

In a German main clause, the conjugated verb must be in position 2.
When you put Den Restmüll first, the subject ich must move behind the verb:

• Normal: Ich bringe den Restmüll raus.
• With fronting: Den Restmüll bringe ich raus.

Because the verb is rausbringen (to take out / bring out), which is a separable verb. In main clauses, the prefix goes to the end:

• Ich bringe den Müll raus. In subordinate clauses, it usually stays together:
• …weil ich den Müll rausbringe.

They’re closely related:

• rausbringen is very common in speech and informal writing (short for herausbringen).
• herausbringen can sound a bit more formal/explicit.

For taking out trash, rausbringen is the most everyday choice.

Restmüll is “residual waste” / “general trash”: the stuff that doesn’t go into recycling categories like paper, bio-waste, packaging, etc.
In many German-speaking places, trash is sorted, so Restmüll is a specific category, not just “any garbage.”

Yes, Tonne literally can mean a large container, but in everyday German it very commonly means a trash bin (especially the outdoor wheelie bin):

• die Mülltonne = the trash bin
  Often people shorten it to die Tonne when it’s clear from context.

nur einmal pro Woche = “only once per week.”
pro (“per”) is common and neutral in German, especially with rates/frequencies:

• zweimal pro Tag = twice per day
  Alternatives:
• einmal in der Woche (very common)
• einmal die Woche (more colloquial)

Because weil introduces a subordinate clause, and German normally separates subordinate clauses with a comma:

• main clause, weil
  • subordinate clause

This comma is standard in both formal and informal writing.

In a weil-clause (subordinate clause), the conjugated verb goes to the end:

• …weil die Tonne sonst zu schnell voll wird.

So the structure is: subject + other parts + verb-at-the-end.

Here sonst means otherwise / if not:

• “I only take it out once a week, because otherwise the bin gets full too quickly.”

It signals a consequence if you don’t follow the stated action/frequency.

voll werden means to become full (a change of state), so it uses werden:

• Die Tonne wird voll. = The bin gets/begins to be full.

voll sein would mean it’s already full (a state):

• Die Tonne ist voll. = The bin is full.

Here the idea is: it would fill up too quickly, so wird fits best.