Wiki ijayo, nitarudisha vitabu hivi maktabani.

It literally means the coming week or the week that is coming. Swahili often uses this pattern to express next:

• wiki ijayo = next week
• mwezi ujao = next month
• mwaka ujao = next year

So instead of a single word exactly like English next, Swahili often uses a form meaning coming or following.

Because in Swahili, descriptive words usually come after the noun, not before it.

So:

• wiki ijayo = next week
• vitabu hivi = these books

This is one of the first word-order differences English speakers notice. English says next week and these books, but Swahili usually says the noun first, then the describing word.

Yes. Wiki ijayo and wiki inayokuja can both mean next week.

The difference is mainly style:

• wiki ijayo = shorter, very common
• wiki inayokuja = more literally the week that is coming

Both are natural. Learners will hear both.

Nitarudisha has three main parts:

• ni- = I
• -ta- = will / future tense
• -rudisha = return something, bring back

So nitarudisha means I will return or I will bring back.

This kind of verb packing is very normal in Swahili. A lot of information that English expresses with separate words gets built into one verb.

Because -rudi and -rudisha do different jobs.

• -rudi = to return / come back oneself
• -rudisha = to return something / take something back

Compare:

• Nitarudi kesho. = I will come back tomorrow.
• Nitarudisha vitabu. = I will return the books.

So in your sentence, the speaker is not saying I will come back. They are saying I will return these books.

Because demonstratives like this and these normally come after the noun in Swahili.

So:

• kitabu hiki = this book
• vitabu hivi = these books

That is the normal pattern. English puts this/these before the noun, but Swahili usually puts them after it.

Because hivi has to agree with vitabu.

Swahili nouns belong to noun classes, and words connected to them often have to match that class.

Here:

• kitabu = book
• vitabu = books

Since vitabu is plural, the demonstrative must also be the matching plural form:

• kitabu hiki = this book
• vitabu hivi = these books

So hivi is used because it matches the noun class of vitabu.

Because Swahili does not have articles like English a, an, and the.

Whether something is definite or not is usually understood from context, or from words like this, that, these, and those.

So:

• vitabu can mean books or the books, depending on context
• vitabu hivi clearly means these books

English needs the much more often than Swahili does.

Maktaba means library.
The ending -ni is a locative ending, which often gives the sense of:

• in
• at
• to

So maktabani means something like in the library, at the library, or to the library, depending on context.

In this sentence, because the verb is return, the natural meaning is to the library.

Because the locative ending -ni already helps express place or direction.

So instead of saying something exactly like to the library with a separate preposition, Swahili can simply use:

• maktabani

With this verb, that naturally gets understood as to the library.

You may also hear other location expressions in Swahili, such as:

• kwenye maktaba
• katika maktaba

But maktabani is perfectly normal and compact.

Yes, to some extent.

The time expression wiki ijayo is at the beginning here because that sounds natural and gives it a little emphasis. But you could also say:

Nitarudisha vitabu hivi maktabani wiki ijayo.

That is also understandable and natural.

What matters most is keeping the noun phrases together clearly:

• vitabu hivi
• wiki ijayo

So Swahili has some flexibility, but the internal order inside those phrases usually stays the same.