English doesn't really have an analogue for Japanese particles. As far as I know, no Western languages does.

They're just a Japanese grammar thing you have to learn. Particles are like "markers" that help you to distinguish where one word/phrase of the sentence ends and the next begins and also help to clarify what purpose in the sentence each word/phrase is serving. There are a lot of different particles in Japanese (64 according to Wikipedia).

For example, the は in "わたしは..." that you might have seen earlier in this course is a particle. は is the the "topic marker" particle. If you explicitly mention the topic in your sentence, you place a は after it so people know that's the topic. Whenever you see は being used as a particle like this, you can read the sentence as, "As for , ..." if that helps. For example, わたしは... = "As for me..."; ねこは... = "As for the cat..."; etc.

Another particle is が in sentences like パンが好きです (I like bread). This one is the identifier particle in this context. It functions similarly to は and many people get the two confused. One of the main distinctions is that は emphasizes what comes after it, while が emphasizes what comes before it. You can think of が as if it were were marking the answer to a question and you were emphasizing the answer.

For example, consider the sentences たなかはせんせいです and たなかがせんせいです. They both say "Tanaka is a teacher" and are grammatically correct sentences, but you'd use them in different contexts. If someone asked "What's Tanaka's job?", you'd use the sentence with は in it, because you want to emphasize the latter part of the sentence - "Tanaka is a teacher." But if you were in a room of people and someone asked, "Who is a teacher here?", you'd use the sentence with が, because you want to emphasize the first part of the sentence - "Tanaka is a teacher."

Anyway, back to the original sentence: に in this context is a particle being used to mark the time that something is happening. The closest English translation would be "at." 九時ごろに = "At around 9:00.."

Later in the sentence, へ is also a particle. It's used to mark the direction the person is going. The closest English translation would be "to" or "towards." (The closer translation is "towards," but it's also used as "to" often, for reasons I'll explain below.) かいしゃへ行きます = "Going to/towards the office."

To confuse you a bit more, に can also be used as a particle to indicate direction and translated as "to." に and へ are largely interchangeable when deciding which direction particle to use, with the subtle difference being that に is closer to a literal "to" and へ is more of "towards" (aka "in the direction of"). However, they are not interchangeable to mark the time something happens - you can only use に for that. That's why へ is often used in the sentences in these lessons to mean "to" even though the more accurate translation is "towards" - because に is already being used earlier in the sentence as the particle to mark the time. Technically, you can use the same particle multiple times and it wouldn't be incorrect, but Duo just likes to use different particles to keep it cleaner.

To my understanding it just tells you in what context is the sentence or word used in, knowing this you can then tell that the sentence is talking about time. Dont take my word for it

Yes it can. Don't quote me on this because I've read it here in the comments. If you use ni you are emphasizing the location. If you use he you are emphasizing the direction. Additionally, you would use ni to say that you're going there and do something like study. If you use he you're just saying you're going towards the school, maybe to meet someone there and go from there not doing anything in particular at that location.

The simple answer is: because it's "at around nine o'clock" instead of "around at nine o'clock".

The more complicated answer is: because ごろ is an adverbial noun suffix in Japanese and as a suffix, it can't very well exist on its own. Inserting a particle after 時 here essentially cuts the entire structure in half - it links 時 to the verb, without ごろ, and ごろ becomes just a lonely suffix word with no noun to modify, so the entire sentence structure just falls apart.