This has to do with the way thought is integrally related to ego. I’m using “ego” here to mean the narrative of self, the “I” that is assumed to be at the receiving end of experience. That narrative is itself the result of a thought process, and most thought features this character in the role of narrator. The sense of “me” is right there embedded in the fabric of thought, and it is thought that sustains the sense of “me”. That self seems like a constant presence, because whenever thought arises the sense of self is right there as an element of the syntax of thought (which is the illusion that led Descartes to his cogito).
But experience is not necessarily mediated by thought. Yes, thought does process experience after the fact, and in doing so it writes in this self character as the receiver of the experience. That thought about the experience tells a story of how I experienced such-and-such, and that’s the way the whole thing gets archived in memory. But that’s not the nature of experience as it happens. At the moment experience unfolds, before it’s run through the thought process, there is no division between a supposed “me” and the object of my experience. There’s just the experience. Strictly speaking, there is not yet an I who experienced it, nor a thing that was experienced.
Many forms of meditation direct the attention to this moment of experience as it unfolds. Still, the attentive absorption is rarely so complete that the brain doesn’t continue its habitual processing of the experience into thought form. Now, here’s the important point as far as your question is concerned: the arising of these thoughts is just as much an aspect of the moment-by-moment unfolding of experience as anything else. The narrative that they contain, however, has nothing to do this experience.
A good analogy to this might be the scrolling feed that the news networks roll across the bottom of their newscasts. You could legitimately say that this scrolling text is part of the newscast, and that those words racing by are just as much a part of the experience of watching CNN as the images on the rest of the screen. But those words are often about something quite different than the rest, and if the attention gets absorbed in the content of the scroll, it will end up on a different track. It’s certainly possible to be aware of the scrolling by of the text without getting caught up in its contents.
Thoughts are like the scrolling text. They’re pretty much always streaming by. Sometimes—often, even—it’s worthwhile to engage them with the attention. But it’s also worth understanding that thought represents a reality remade according to the ground rules of thought itself.
To say that “I observe my thoughts go by like bubbles” is not accurate. That’s already a product of the interpretative spin applied by the thought process. In the actual experience this is attempting to describe, there is no aloof observer, nor anything that’s observed. This kind of wording has led you to think that there’s a “me” having a thought and another “me” off at a distance watching that happen, but that’s not how it is. That’s just another thought created by the compulsion to put a “me” at the root of experience.