Learn Hardcore German: Im Tierheim arbeiten viele Leute, die sich um einsame Tiere kümmern. - In the animal shelter many people work who take care of lonely animals.

in + dem → im (in the)
This contraction is very common and sounds more natural in everyday German.
You can say in dem Tierheim, but it sounds more formal or emphasized. In most contexts, im Tierheim is preferred.

Why is Tierheim in the dative case here?

Because in can take either accusative or dative, and here it’s used with dative to express location (answering “Where?”).

Wo? (Where?) → dative
- Im Tierheim arbeiten viele Leute. (They work in the shelter – location)
Wohin? (Where to?) → accusative
- Sie gehen ins Tierheim. (= in das Tierheim, movement to the shelter)

So im Tierheim = in dem Tierheim (dative singular neuter).

Why is the word order Im Tierheim arbeiten viele Leute and not Im Tierheim viele Leute arbeiten?

German main clauses follow the verb-second rule: the conjugated verb must be in second position.

You can move different elements to the first position for emphasis, but the verb still has to come second:

Viele Leute arbeiten im Tierheim. (subject first)
Im Tierheim arbeiten viele Leute. (place first → verb comes next)

Im Tierheim viele Leute arbeiten is wrong in a main clause, because the verb is not in the second position.

What exactly does Leute mean, and can I say ein Leute?

Leute means people in a general, informal sense.

It is plural only. There is no singular form like ein Leute.
Instead, for one person you say eine Person, ein Mensch, ein Mann, eine Frau, etc.

You can often replace Leute with Menschen, but Leute is more colloquial:

Viele Leute arbeiten im Tierheim. (very normal)
Viele Menschen arbeiten im Tierheim. (slightly more neutral/formal)

Why is the relative pronoun die used after Leute?

The relative pronoun must agree with its antecedent (here: Leute) in gender and number, and its case comes from its function in the relative clause.

Leute is grammatically plural (and “people” don’t really have a natural gender in German), so the relative pronoun is die (plural nominative).
In the relative clause die sich um einsame Tiere kümmern, die is the subject of kümmern, so it’s nominative plural.

So you get:

viele Leute, die sich um einsame Tiere kümmern
= many people who take care of lonely animals

What is the function of sich in die sich um einsame Tiere kümmern? Why is it reflexive?

Sich kümmern um is a reflexive verb in German. You must use a reflexive pronoun with it:

ich kümmere mich
du kümmerst dich
er/sie/es kümmert sich
wir kümmern uns
ihr kümmert euch
sie/Sie kümmern sich

In the sentence, the subject of the relative clause is die (referring to Leute), so the corresponding reflexive pronoun is sich (3rd person plural or formal).

You cannot omit sich here; kümmern um without sich is not standard German.

Why is the preposition um used with sich kümmern? Could it be für instead?

With sich kümmern, German normally uses um + accusative:

sich um etwas / jemanden kümmern = to take care of / look after something or someone.

So:

sich um einsame Tiere kümmern
literally: “to take care around lonely animals” → idiomatically: “to take care of lonely animals”

Um here is just part of the verb’s normal pattern and always takes the accusative.

Using für with kümmern (sich für etwas kümmern) is either wrong or sounds very old-fashioned/dialectal in modern standard German.

Why is einsame Tiere in the accusative case?

Because the preposition um always takes the accusative.

The structure is:

sich um + Akkusativ kümmern

So:

um einsame Tiere → Tiere is accusative plural.
You could also see it if you replace it with a pronoun:
Sie kümmern sich um sie.
(Here sie is accusative plural = them.)

Why does the adjective end in -e in einsame Tiere and not -en or something else?

The form einsame comes from adjective declension rules:

Tiere is plural accusative.
There is no article (no die, meine, etc.) before the adjective.
In plural, without any article, the adjective takes -e in nominative and accusative.

Patterns (strong declension, no article):

nominative plural: einsame Tiere
accusative plural: einsame Tiere

So einsame is the correct ending here.

Why is there no article before einsame Tiere? Why not die einsamen Tiere?

Leaving out the article makes the phrase more general and unspecific:

einsame Tiere = lonely animals in general (not specific ones).
die einsamen Tiere = the lonely animals (some specific group that you and the listener know about).

In a sentence about typical work in an animal shelter, the speaker likely means any lonely animals that are there, not some particular, already-defined group, so the article is omitted.

Why is the verb kümmern at the end of die sich um einsame Tiere kümmern?

Because this is a relative clause, and in German, in most subordinate clauses the conjugated verb goes to the end.

Compare:

Main clause: Sie kümmern sich um einsame Tiere. (verb in 2nd position)
Relative clause: … die sich um einsame Tiere kümmern. (verb at the end)

The comma before die marks the beginning of the subordinate clause, and from that point the word order changes: the conjugated verb moves to the end.

Can I change the order of the two parts and say: Viele Leute, die sich um einsame Tiere kümmern, arbeiten im Tierheim?

Yes, that sentence is also perfectly correct.

You have two valid versions:

Im Tierheim arbeiten viele Leute, die sich um einsame Tiere kümmern.
Viele Leute, die sich um einsame Tiere kümmern, arbeiten im Tierheim.

Both follow the verb-second rule in the main clause and put the verb at the end in the relative clause. The difference is just what you put in the first position (emphasis): the place (im Tierheim) or the people (viele Leute).