La professora explica aquest tema amb un altre exemple.

Because professora is a feminine noun meaning a female teacher, so it takes the feminine singular article la.

• la professora = the female teacher
• el professor = the male teacher

Catalan, like English, uses the definite article here because we are talking about a specific teacher.

Catalan often leaves out subject pronouns when they are not needed. The verb form already shows the person and number.

• explica = she explains, he explains, or it explains, depending on context

So La professora explica... is completely natural, and adding ella would usually sound emphatic:

• La professora explica... = neutral
• Ella explica... = she explains, with extra emphasis or contrast

Explica is the third-person singular present indicative of explicar.

Here it means:

• explains
• sometimes is explaining, depending on context

So:

• La professora explica... = The teacher explains... / The teacher is explaining...

Because tema is masculine singular, so the demonstrative must also be masculine singular:

• aquest tema = this topic
• aquesta idea = this idea

A useful thing to remember: some Catalan nouns ending in -a are still masculine, especially words of Greek origin, such as:

• el tema
• el problema
• el sistema
• el programa

So even though tema ends in -a, it is masculine.

Because aquest already functions as a demonstrative determiner, like this in English. In standard Catalan, you normally do not put a definite article before it.

• aquest tema = this topic
• not el aquest tema

So aquest does the job that this does in English.

Both are demonstratives, but they point to different degrees of distance.

• aquest = this
• aquell = that

So:

• aquest tema = this topic
• aquell tema = that topic

In a literal physical sense, aquest is for something closer. In a more abstract sense, it can also mean something more immediate in the conversation or context.

Because un altre is the normal way to say another or one more.

• altre by itself means other
• un altre means another

So:

• un altre exemple = another example
• l’altre exemple = the other example

Also, exemple is masculine singular, so you need:

• un altre exemple

If the noun were feminine, it would be:

• una altra idea = another idea

Amb usually means with.

In this sentence, it shows the means or method:

• amb un altre exemple = with another example / by using another example

So the idea is that the teacher explains the topic using another example.

Literally yes, but in natural English the idea is often better expressed as:

• by using another example
• through another example

In Catalan, amb is very common for this kind of meaning. So the sentence suggests that the teacher explains the topic by means of an additional example.

The sentence uses the most neutral word order:

• La professora = subject
• explica = verb
• aquest tema = direct object
• amb un altre exemple = prepositional phrase

Catalan word order is more flexible than English word order, so other versions are possible, especially for emphasis:

• Amb un altre exemple, la professora explica aquest tema.

That is still correct, but the original sentence is the most straightforward and neutral version.

A rough pronunciation guide is:

luh pruh-fuh-SOR-uh eks-PLEE-kuh uh-KEST TEM-uh am oon AL-truh eg-ZEM-pluh

A few useful notes:

• ll in la is just like English l
• professora has the stress on -so-
• explica has the stress on -pli-
• aquest has the stress on -quest
• exemple has the stress on -zem-

Depending on the Catalan variety, some vowels may sound a little different, but this will get you close.