Learn Hardcore Japanese: sekaizyuu de nihongo wo benkyousuru hito ga huete imasu. - The number of people studying Japanese is increasing all over the world.

If you said 世界中に日本語を勉強する人がいる, that would mean
There are people who study Japanese all over the world (existence),
while 世界中で〜勉強する人が増えています focuses on the activity of studying happening worldwide.

Why do we say 日本語を勉強する and not just 日本語勉強する?

Because 勉強する is a verb that takes a direct object, and that object is marked with を.

日本語 = Japanese (language)
日本語を勉強する = to study Japanese

So を marks what is being studied.

Note: You might also see:

日本語を勉強する
日本語の勉強をする

Both are natural. The first treats 勉強する as a verb “to study”;
the second treats 勉強 as a noun “study,” and 〜をする makes it a verb.
In your sentence, we’re using the simpler 日本語を勉強する pattern.

How does 日本語を勉強する人 work grammatically? Why is the verb before 人?

日本語を勉強する人 is a relative clause:

日本語を勉強する = (someone) studies Japanese
人 = person

Put together:

日本語を勉強する人
= the person/people who study Japanese

In Japanese, the describing part comes before the noun it modifies.
So what in English would be:

“people who study Japanese”

becomes in Japanese:

日本語を勉強する人

No relative pronoun like who/that is needed; the verb phrase directly modifies 人.

Why is が used before 増えて instead of は?

が here marks the grammatical subject of the verb 増えている.

Subject: 日本語を勉強する人 (people who study Japanese)
Verb: 増えています (are increasing)

So:

日本語を勉強する人が増えています。
= The people who study Japanese are increasing
(i.e., the number of such people is increasing)

Could you use は?

日本語を勉強する人は増えています。

This is possible, but は then makes that phrase the topic, often with some contrast or known-background nuance, like:

“As for people who study Japanese, they are increasing (compared to before / compared to others).”

With が, it’s a neutral statement identifying that group as the subject of “increasing.”

What exactly does 増えています mean? Is it “are increasing” or “have increased”?

増えている is 増える (to increase) in the -ている form.

For many intransitive change verbs like 増える, -ている can express:

A current ongoing change
- are increasing / are on the rise
A resulting state after a change
- have increased (and are now many)

Context decides which nuance is stronger.
In this sentence, it naturally reads as:

The number of people who study Japanese *has been increasing / is on the rise (and is now higher than before).*

So it covers both the idea of process and the current state resulting from that process.

Why is there no explicit word for “number” like 数? How can “people are increasing” make sense?

Japanese often says “people are increasing” where English says “the number of people is increasing.”

日本語を勉強する人が増えています。
Literally: People who study Japanese are increasing.
Interpreted as: The number of people who study Japanese is increasing.

If you want to say it more literally with “number,” you can say:

世界中で日本語を勉強する人の数が増えています。
= The number of people who study Japanese around the world is increasing.

Both are correct; the original is just more natural and concise in Japanese.

Could we say 日本語を勉強している人 instead of 日本語を勉強する人? What’s the difference?

Both are grammatically correct, but the nuance differs.

日本語を勉強する人
- Describes someone in a general, habitual sense:
  people who (generally) study Japanese / learners of Japanese
- Suitable for talking about a category or general trend, which matches your sentence.
日本語を勉強している人
- Focuses more on people who are currently in the state of studying Japanese (they’re in the middle of learning it now).
- Sounds a bit more like “people who are (now) studying Japanese”.

In your sentence about a global trend, 勉強する人 (generic learners) is more typical, but 勉強している人 would not be wrong; it just slightly shifts the focus to people actively studying at this time.

Could the word order be changed, like 日本語を世界中で勉強する人が増えています? Is that okay?

Yes, that word order is possible:

世界中で日本語を勉強する人が増えています。
日本語を世界中で勉強する人が増えています。

Both are grammatical.

The difference is focus and naturalness:

世界中で日本語を勉強する人が〜
Starts with 世界中で, putting early emphasis on “all over the world.”
Very natural for stating a global trend.
日本語を世界中で勉強する人が〜
Starts with 日本語を, putting a bit more early focus on the object (Japanese).
Still okay, but slightly less “default” sounding for this kind of statistic-style sentence.

Japanese word order is flexible, but the original is the most typical-sounding arrangement here.

What’s the level of politeness of 増えています? How would plain form look?

増えています is in the polite form because of います (the polite form of いる).

Polite: 増えています。
Plain (casual): 増えている。

The rest of the sentence is also in neutral/polite style, so the full plain version would be:

世界中で日本語を勉強する人が増えている。

Use 増えています in conversation with teachers, in news, writing, etc.
Use 増えている in casual speech with friends, in informal writing, or inner monologue.

What is the full breakdown of the sentence into units and meanings?

世界中で
- 世界中 = all over the world
- で = at/in (place where an action occurs)
  → in/throughout the whole world
日本語を
- 日本語 = Japanese (language)
- を = object marker
  → Japanese (as the thing studied)
勉強する
- 勉強する = to study
  → study
人が
- 人 = person/people
- が = subject marker
  → people who … (subject of the verb that follows)
増えて
- 増える = to increase
- 〜て = connective form to link to いる
  → have increased / are increasing (as a change)
います。
- いる (polite います) = to exist / to be (here, auxiliary for ongoing/resulting state)

Altogether:
世界中で日本語を勉強する人が増えています。
= The number of people who study Japanese around the world is increasing.