Learn Farsi: کتاب روی میز است، من می‌خوانمش. - The book is on the table; I am reading it.

The 'it' is the ending -ash attached to the very end of the verb. The base verb here is می‌خوانم (I am reading). When you attach the object enclitic -ash (meaning him/her/it), it becomes می‌خوانمش (I am reading it).

How do I pronounce the ending of 'می‌خوانمش'?

In formal or written Farsi, the attached object pronoun is pronounced -ash (mikhanamash). However, in everyday spoken Farsi, this ending almost always changes to -esh (mikhanamesh). You will hear -esh much more often in conversations.

Why do I hear a 'ye' sound between 'روی' (on) and 'میز' (table)?

That is the ezâfe! It connects a position word like روی (on/top) to the noun that follows it. Because روی (ru) ends in a vowel sound, the connecting ezâfe sounds like 'ye', making it ru-ye miz. As usual in Farsi, this connecting sound is not written out as a separate letter.

Is the word 'من' (I) strictly required before the verb?

No, you can comfortably drop it. The verb می‌خوانم already contains the personal ending -am, which means 'I'. Including من just adds a little emphasis, but saying the sentence without it is perfectly correct and very natural.