I juni og juli er byen fuld af mennesker, som tager billeder i centrum.

Yes. Here i means in, so I juni og juli means In June and July.

A few useful points:

• Danish normally uses i with months:
  
  • i juni = in June
  • i juli = in July
• Danish does not usually use an article before month names, just like English:
  • i juni, not i den juni

So this opening is a time expression placed first in the sentence.

This is because Danish follows the V2 rule in main clauses. V2 means the finite verb usually comes in the second position.

The sentence is structured like this:

• I juni og juli = first element
• er = verb in second position
• byen = subject

So:

• I juni og juli er byen fuld af mennesker.

If you started with the subject instead, you would get:

• Byen er fuld af mennesker i juni og juli.

Both are correct, but the first version emphasizes the time phrase.

Fuld af means full of.

So:

• byen er fuld af mennesker = the city is full of people

This is a fixed expression in Danish:

• fuld af vand = full of water
• fuld af idéer = full of ideas
• fuld af turister = full of tourists

English uses of, and Danish uses af in the same kind of structure.

Mennesker is the indefinite plural form: people.

• mennesker = people / human beings
• menneskene = the people

Here the sentence is talking about people in a general sense, not a specific known group, so mennesker is the natural choice:

• byen er fuld af mennesker = the city is full of people

If you said menneskene, it would sound more like the people, meaning a specific group already known from context.

Som introduces a relative clause. Here it means who or that:

• mennesker, som tager billeder i centrum
• people who take pictures in the center

It connects mennesker with extra information about them.

So the sentence says that the city is full of people, and then adds which people: the ones who are taking pictures in the center.

Yes. In sentences like this, som often corresponds to English who, which, or that, depending on what it refers to.

Examples:

• en mand, som smiler = a man who is smiling
• en bog, som er dyr = a book that/which is expensive

In your sentence, because it refers to mennesker, English would most naturally use who:

• mennesker, som tager billeder = people who take pictures

Because som tager billeder i centrum is a relative clause, and in standard Danish writing it is separated with a comma.

So:

• mennesker, som tager billeder i centrum

The comma helps show that the second part gives more information about mennesker.

This is more consistent in Danish than in English, where commas before who/that depend more on the type of relative clause.

Literally, tager billeder means takes pictures.

• tage = to take
• billeder = pictures

This works very much like English:

• at tage et billede = to take a picture
• at tage billeder = to take pictures

Even though English also has photograph, Danish very commonly uses tage billeder in everyday speech.

Because this is a general plural object:

• tager billeder = take pictures

Compare:

• tager et billede = takes a picture
• tager billeder = takes pictures
• tager billederne = takes the pictures

So billeder here means pictures in general, not one specific set of pictures.

I centrum means in the center or downtown / in the city center, depending on context.

• centrum = center, central area
• i centrum = in the center / in the city center

In this sentence, it most naturally means in the city center.

You may also see:

• i bymidten = in the town/city center
• i centrum af byen = in the center of the city

Byen is the noun by with the definite singular ending added.

• by = city / town
• byen = the city / the town

Danish usually attaches the definite article to the end of the noun instead of putting a separate word in front:

• en by = a city
• byen = the city

So byen is literally city-the, but in natural English it is simply the city.

Sometimes Danish allows der in relative clauses, especially when it is the subject of the clause, but som is the safest and most standard choice here for learners.

So:

• mennesker, som tager billeder = definitely correct

You may hear or see der in some contexts, but if you are unsure, use som.

Grammatically, som tager billeder i centrum describes mennesker.

So the full idea is:

• the city is full of people who are taking pictures in the center

That means the relative clause limits or identifies which people are being talked about.

If you wanted to say the city is full of people, and separately mention that they take pictures downtown, you would probably phrase it differently.