The problem is, all these sentences are given without context. I'm still learning but here's how I see the difference:
"Zij leest een boek." or "She is reading a book." might be said in response to, "What's Rose doing today?" It's used in a more general sense. "Zij leest boeken" could mean, "She reads books." Like, "Does she have a hobby?" "She reads books."
"Ik ben een boek aan het lezen!" or "I am reading a book!" might be used in response to, "You've been in the bathroom for an hour! What's taking you so long?"
You need "aan het" for the continuous construction. I did not come up with that, so I can't tell you why. It is simply how the Dutch grammar works.
Technically you could say: "Ik ben schrijvende"
In that case, "schrijvende" has the function of an adjective. However, this construction is not used much and sounds very rather awkward.
Have a read through this: https://www.duolingo.com/comment/4298404
Or through this: http://www.dutchgrammar.com/en/?n=Verbs.Ot02
In short, if in Dutch you want to emphasise that something is happening at this moment you can use the "continuous" which is "a conjugation of zijn + aan het + infinitive of the verb_.
- Ik ben aan het lopen - I am walking
- Wij zijn aan het eten - We are eating
- Hij is naar school aan het fietsen - He is cycling to school
- Zij zijn door het bos aan het wandelen - They are walking through the forest.