Learn Hardcore German: Die Bücher sollen heute noch eingepackt werden. - The books are supposed to be packed today.

Because this is a passive sentence. In the passive, the thing that would be the direct object in an active sentence becomes the subject and therefore takes nominative:

Active: Jemand packt die Bücher ein. (die Bücher = accusative object)
Passive: Die Bücher werden eingepackt. (die Bücher = nominative subject)

What does sollen mean here, and why not müssen?

Sollen often expresses that something is expected / supposed / intended—frequently based on a plan, instruction, or someone else’s expectation. It can feel slightly less “hard necessity” than müssen.

Die Bücher sollen … eingepackt werden. = they are supposed to be packed (plan/instruction)
Die Bücher müssen … eingepackt werden. = they must be packed (strong necessity)

In context, sollen can also imply “according to what someone says/has ordered.”

Why are there two verbs at the end: eingepackt werden?

That’s the passive construction:

werden = passive auxiliary (here in the infinitive because of the modal)
eingepackt = past participle of einpacken

So eingepackt werden = to be packed.

Why is werden at the very end of the sentence?

Because German puts non-finite verb parts (infinitives/participles) in a verb cluster at the end, especially with modals:

sollen is the finite verb in position 2 (main-clause rule)
the rest (… eingepackt werden) goes to the end

Structure: Subject – sollen – (time adverbs) – Participle – werden

How do I know this is passive and not future tense with werden?

Future with werden uses an infinitive of the main verb:

Future: Die Bücher werden eingepackt. (could mean “will be packed,” but often interpreted as present/future without clear context) Passive here is made unmistakable by:
past participle (eingepackt) + werden And with a modal it’s clearly passive:
sollen … eingepackt werden = supposed to be packed

What tense is sollen in here?

It’s present tense: sollen (3rd person plural).
German often uses the present tense with time words like heute to refer to a future plan:

heute noch makes it clear it’s intended later today (still today).

What exactly does heute noch mean, and why is noch there?

heute noch means still today / before today is over.
noch adds the idea that there is remaining time in the day and the action should happen within that remaining time:

Die Bücher sollen heute noch … = the books are supposed to be packed later today at the latest.

Why is it eingepackt and not einpackt?

Because in the passive you use the past participle of the main verb:

infinitive: einpacken
participle: eingepackt

The ge- is inserted after the prefix ein-, giving ein + ge + packt → eingepackt.

Can this sentence mention who is packing the books?

Yes. You can add an agent with von (often for people/institutions) or durch (often for means/process):

Die Bücher sollen heute noch von den Mitarbeitern eingepackt werden.
… durch eine Maschine eingepackt werden.

If you don’t mention it, the sentence stays agentless, which is very common in German passive.

Could I also say it in active voice, and would it feel different?

Yes, and it often feels more direct because it names or implies an agent:

Active (generic): Man soll die Bücher heute noch einpacken.
Active (specific): Wir sollen die Bücher heute noch einpacken.

The passive version (Die Bücher sollen … eingepackt werden) keeps the focus on the books rather than on who is doing the packing.

What happens to the word order in a subordinate clause?

In a subordinate clause, the finite verb goes to the end, so sollen moves into the final verb cluster:

…, weil die Bücher heute noch eingepackt werden sollen.

You’ll often see this exact cluster: eingepackt werden sollen.