Mit dem Kofferanhänger finde ich meinen Koffer an der Gepäckausgabe.

German often puts something other than the subject in the first position for emphasis or style.

Here, Mit dem Kofferanhänger is in the first slot, so the conjugated verb must still stay in second position:

• Mit dem Kofferanhänger finde ich meinen Koffer ...
• Ich finde meinen Koffer mit dem Kofferanhänger ...

Both are grammatical, but the original sentence emphasizes how I find the suitcase.

Because mit always takes the dative case.

• der Kofferanhänger = the luggage tag
• masculine singular in the dative = dem Kofferanhänger

So:

• mit dem Kofferanhänger = with the luggage tag

In German compound nouns, the last part usually determines the gender.

• der Anhänger = tag, label, pendant, trailer (depending on context)
• so der Kofferanhänger is also masculine

That is why you get:

• nominative: der Kofferanhänger
• dative after mit: dem Kofferanhänger

Kofferanhänger is a compound noun:

• Koffer = suitcase
• Anhänger = tag / label / thing attached to something

So here it means a luggage tag or baggage tag, especially the one that helps you identify your suitcase.

Because meinen Koffer is the direct object of finden, and Koffer is masculine.

The verb finden takes an accusative object:

• nominative: mein Koffer
• accusative: meinen Koffer

So:

• Ich finde mein Koffer ❌
• Ich finde meinen Koffer ✅

Because German main clauses follow the verb-second rule.

The first position is taken by Mit dem Kofferanhänger, so the verb comes next:

1. Mit dem Kofferanhänger
2. finde
3. ich
4. meinen Koffer
5. an der Gepäckausgabe

If the subject comes first, then the order is more like English:

• Ich finde meinen Koffer an der Gepäckausgabe mit dem Kofferanhänger.

Here an means something like at in the sense of at the baggage claim area.

The preposition an can take either accusative or dative:

• dative for location
• accusative for direction/movement toward something

In this sentence, it is a location, so German uses the dative:

• die Gepäckausgabe = the baggage claim
• dative singular = der Gepäckausgabe

So:

• an der Gepäckausgabe = at the baggage claim

Both prepositions can sometimes be translated as at or in, but German uses different ones depending on the idiomatic expression.

an der Gepäckausgabe is the normal way to say at the baggage claim / at baggage reclaim.

Using in der Gepäckausgabe would sound more like inside the baggage-claim area/building/room, which is not the usual idea here.

So this is mostly a matter of standard German usage.

It can mean both, depending on context.

In this sentence, finden is best understood as:

• find
• locate
• identify

At baggage claim, you are not necessarily discovering a lost suitcase; you are spotting the correct one among many. So in English, identify my suitcase may sound more natural in context.

It is another compound noun:

• Gepäck = luggage / baggage
• Ausgabe = issuing, handing out, distribution

So Gepäckausgabe literally means something like baggage distribution/collection point, which in natural English is baggage claim.

Yes. That version is also very natural.

The difference is small:

• Mit dem Kofferanhänger finde ich meinen Koffer ... = I find my suitcase with the luggage tag.
• Mit dem Kofferanhänger kann ich meinen Koffer ... finden = With the luggage tag, I can find my suitcase.

The version with kann emphasizes ability or helpfulness. The original sentence sounds more direct and factual.

Not exactly.

• der Koffer = a suitcase
• das Gepäck = luggage/baggage in general

So in the sentence:

• meinen Koffer = my specific suitcase
• an der Gepäckausgabe = at the baggage claim area, where luggage is collected

Yes, that is a very good way to understand it.

mit often means with, but in many sentences it also has the sense of:

• by means of
• using

So:

• Mit dem Kofferanhänger finde ich meinen Koffer
  can be understood as
• Using the luggage tag, I find my suitcase

German present tense can do both.

So this sentence can mean:

• I am finding my suitcase with the luggage tag in a present situation, or
• I find my suitcase with the luggage tag as a general fact / usual result

German does not always distinguish these as clearly as English does with I find vs I am finding. Context tells you which meaning is intended.