The first thing we need to address are those awkward brackets/italics/footnotes involving 7:53-8:11 in most newer Bible translations. What does it mean when it says that "The earliest mss do not include" these verse? Why does anybody tinker with the Bible?
To wrestle with this takes a minute or two, so bear with me. As a foundation, I would remind you that I believe the original text of Scripture, in its original languages, is divinely inspired and without any mixture of error in its content. That is, what John wrote in the language John wrote it is inerrant. We start there.
Now, the challenge comes as we examine what the original text actually was. We have no "originals" (known as "autographs") in our possession. What we have are copies. Lots of copies, to be certain, with more copies of the New Testament than of any other ancient writing of similar age. These are in the form of full copies, partial copies, comments by ancient Christians, and quotations. They exist in Greek, Latin, Aramaic, Syriac, and several other languages.
This results in a form of study called "textual criticism," which strives to examine the available texts (called "extant" texts) and determine what the original text was. And, for the most part, there is somewhere between 95 and 99% agreement with each other. (Read Daniel Wallace's Revisiting the Corruption of the New Testament for a full examination of the issue.)
Now, we get to our text, a problem arises that is hard to sort out. When one looks at the copies of John from the first several centuries, John 7:53-8:11 is not there. (see here: https://danielbwallace.com/2013/06/26/where-is-the-story-of-the-woman-caught-in-adultery-really-from/ )It's also not cited by ministers of the Early Church. John Chrysostom's sermons, for example, go sequentially through John...and go straight from John 7:52 to John 8:12. But, the passage shows up in other copies in other Gospels, and then it turns up in John in the Greek texts used for the King James and other early English versions. (It's also in the Latin Vulgate, which dates back hundreds of years earlier.)
So it seems like it could belong, but there is some uncertainty. We can see the character of Jesus clearly here, but there is nothing new that we learn. So there is no theological reason to set it aside. That's why most Bible translations try to hit the middle ground and note the textual problem but maintain the passage. Personally, I'm in favor of leaving it in, but I'm a pastor more than a scholar and I see how carving out passages like this can undercut our understanding of God's Word. If we are not careful, we start moving Scripture out of the hands of God's people and back to an "experts-only" situation. That's not healthy for the church.
The rest of the passage deals with Jesus pointing out the difference in what the religious leaders taught and what God had said and what God was saying, right there, through Him. The wrap-up of John 8 has Him reminding the Jewish leaders (usually, John uses the term "the Jews" for the leaders, not all of the Jews; some folks misconstrue this into anti-Semitism) that He pre-existed Abraham. Putting it is "I am" pretty well showed them what He meant.
Chapter 9 then opens with the healing of the man born blind. Jesus points out that his blindness is for the glory of God. More on him in the next section.