Mambo rafiki! This is not still really clear to me, and I am working hard on it. So, I will just try to show some examples to help (I see you want to know if every verb form shows this derivation rule, and I want to know it too):
(I am afraid they are not as simple sentences as in the course. The following text in the link is all about verb derivation, and it is long)
Some sentences with prefixes and infixes:
Aliniandikia barua, he wrote a letter to me, or on behalf of me.
Tutamsomea kitabu, we will read a book to her.
Notice that the above sentences have two objects: the direct (barua, kitabu), and the indirect (mimi, yeye), the ones for whom the actions were done. In a prepositional verb form, it is ALWAYS the indirect object which is included in the verb as an object prefix.
Compare these sentences:
Alisoma barua? Aliisoma. Did he read the letter? He read it.
Alikusomea barua? Alinisomea. Did he read you the letter? He read it to me.
In the first sentence, barua is a direct object, and included as object prefix in the reply. In the second sentence, barua is also the direct object, but it is not included as an object prefix. The object prefix in the verb is wewe (KU) in the question, mimi (NI) in the reply - the indirect object.
From KamusiGOLD (this site seems to be under reconstruction):
This text below presents more example sentences (Object Infixes for Things):
As you learned in the last unit, object infixes are placed between the tense marker and the verb stem:
Ulikinunua wapi kitanda hiki? Where did you buy (it) the bed?
Nilikinunua Kenya. I bought it in Kenya.
As you learned in the last unit and can see in the first example above, it’s possible to use an object infix even if the noun itself appears in the sentence. This adds more emphasis to that object. If the object is already known from context, the object infix alone may be used, just as is the case with English direct object pronouns. Here are a few more examples, with the noun class indicated to help you pick out the relevant infixes.
Kitabu hiki nilikisoma shule. I read this book at school.
Vitabu hivi nilivisoma shule. I read these books at school.
Fundi anaiangalia kompyuta. The technician is checking the computer.
Kompyuta zangu nitaziwacha hapa. I’ll leave my computers here.
There’s just one more point to make about object infixes in general. The infixes you learned in Unit 13 are the ones you use for people, so they are the ones that correspond to M-Wa nouns. However, as you know, there are several examples of nouns referring to people that belong to other noun classes: baba (father/s) or askari (soldier/s), from the N- Class, daktari/madaktari (doctor/s), from the Ji-Ma class, and so on. Just as with other types of agreement, these nouns denoting human beings use M-Wa object infixes, even though they’re technically in other classes:
Ulimwona daktari leo asubuhi? Did you see the doctor this morning?
Ndio, nilimwona. Yes, I saw him.
From Living Language | Advanced Swahili - Lesson 15: http://www.livinglanguage.com/languagelab/swahili/11258/advanced-grammar
The problem I am having now is that in the first sentences we learned , there were no object infixes such as "i am reading the book" Ninasoma kitabu.....but now we are learning to put in an object infix so it is now "ninakisoma kitabu".... so do I need the obbject infix if it is just an object as the direct object or can I leave it out like were learned in the first place?
Aha! I see now, in this PDF it says that if the object is animate then you have to use an object infix, if it's inanimate you can use just the object or both the infix and the object (and of course just the infix if the object is known) :) https://www.google.com/url?sa=t&source=web&rct=j&url=https://www2.ku.edu/~kiswahili/pdfs/lesson_31.pdf&ved=0ahUKEwiN7f6kkenSAhWBOyYKHTVyDcEQFggaMAA&usg=AFQjCNFctaQQKt8BppFnmw-3jpneiDosiQ&sig2=XH-JlZlVnx2AMTxfDzk_rQ
I will only attempt to answer your last question which asks "Why is the 'them' (-wa-) needed?". I will change this question to "Why do we need object infixes in general?" and then try to answer that instead.
Object infixes help us understand something about the object receiving the action of the verb. That something is the class of the object receiving the action. Let's take "walisalimu" and "waliwasalimu" for example. From "walisalimu", all we know is that "they greeted" but we don't know who they greeted. On the other hand, if we say "waliwasalimu" we know that they greeted an object of plural form in the m-/wa- noun class. And , that object happens to be wazazi in our example. So, we have this extra bit of information because of the object infix -wa- in our particular example.
To understand which object infix to use for which noun class, refer to the tips and notes page here https://www.duolingo.com/skill/sw/Object-Infixes
Disclaimer: The explanation given above is based on my own understanding, which may be wrong. So use your own judgement. If someone finds errors in my explanation please point them out because I am here to learn.