Bei der Anprobe prüfen wir die Kleidergröße.

Because bei takes the dative case.

The noun Anprobe is feminine, so its article changes like this:

• nominative: die Anprobe
• accusative: die Anprobe
• dative: der Anprobe

So after bei, you get bei der Anprobe.

In this sentence, bei der Anprobe means something like during the fitting or at the fitting.

Here, bei does not literally mean only by. In this kind of sentence, it often means:

• during
• at
• on the occasion of

So Bei der Anprobe prüfen wir die Kleidergröße means that the checking happens during the fitting.

This is a very common use of bei in German.

German main clauses usually follow the verb-second rule.

That means the finite verb must come in position 2. In this sentence:

• position 1: Bei der Anprobe
• position 2: prüfen
• then: wir die Kleidergröße

So even though the sentence starts with a prepositional phrase, the verb still has to come second.

Because the sentence does not start with the subject.

German lets you put different elements first for emphasis or structure. Here, Bei der Anprobe is placed first. Once that happens, the verb must come next, and the subject follows it:

• Bei der Anprobe
  • prüfen
    • wir
      • die Kleidergröße

If you started with the subject, you would get:

• Wir prüfen bei der Anprobe die Kleidergröße.

That is also correct, but the original sentence emphasizes the situation or time: during the fitting.

Anprobe is a noun meaning fitting or trying on clothes.

It comes from the verb anprobieren, which means to try on.

So:

• anprobieren = to try on
• die Anprobe = the fitting / the try-on session

In many contexts, Anprobe suggests a situation where someone tries on clothes so that fit, size, or appearance can be checked.

They are related, but they are different parts of speech:

• die Anprobe = a noun
• anprobieren = a verb

Examples:

• Bei der Anprobe prüfen wir die Kleidergröße.
  Here, Anprobe is the event or situation: the fitting.

• Wir probieren das Kleid an. or Wir anprobieren das Kleid.
  Here, the verb means to try on the dress.

So the sentence is not saying we try on the size. It is saying that during the fitting, we check the clothing size.

Kleidergröße is a compound noun:

• Kleider = clothes / garments
• Größe = size

Together, Kleidergröße means clothing size or dress size, depending on context.

In German, compound nouns are very common. The last part is the main noun, so:

• Größe is the head noun
• therefore the whole word is feminine: die Kleidergröße

That is why the sentence says die Kleidergröße.

Because Kleidergröße is a feminine noun.

The head of the compound is Größe, and Größe is feminine: die Größe. That makes the whole compound feminine too: die Kleidergröße.

Also, in this sentence Kleidergröße is the direct object of prüfen, so it is in the accusative case. For feminine nouns, the accusative article is still die:

• nominative: die Kleidergröße
• accusative: die Kleidergröße

So die is exactly what you expect here.

Yes, absolutely.

Both are correct:

• Bei der Anprobe prüfen wir die Kleidergröße.
• Wir prüfen bei der Anprobe die Kleidergröße.

The difference is mainly one of focus or style.

• Starting with Bei der Anprobe emphasizes when/in what situation the checking happens.
• Starting with Wir is more neutral and starts with the subject.

This kind of word order flexibility is normal in German, as long as the verb stays in second position in a main clause.

Usually, yes. Prüfen often means:

• to check
• to examine
• to verify

In this sentence, prüfen wir die Kleidergröße means we check the clothing size.

It sounds a bit more formal or careful than a very casual English look at. It suggests that the size is being verified as part of the fitting process.