One of the effects of meditation is that thoughts lose their "significant" status. Usually we are embedded in our thoughts (and the feelings they evoke). Your thoughts and feeling pull strongly on your attention, drawing you into an inner world, somewhat removed from the outer world. We, in a sense, live in our thoughts. They strongly color our experience.
When you meditate, you continually interrupt your own thoughts. You dismiss your thoughts and go back to the mantra or your breath. Slowly but surely, your thoughts lose their status. Instead of the natural rulers, thoughts become annoying pests.
This change in your relationship with your own thoughts is an important side-effect of meditation, and is probably responsible for many of meditation's benefits. This new, lower status of your thoughts allows you to gain a kind of detachment from the drama of your daily life — the drama that normally captures and enmeshes you so much. It allows you to maintain your inner peace for more and more of your life.
Again and again you get sucked into the thought-world only to realize you've been lost in a daydream, and you drop it and come back to the mantra. Over time, you begin to understand that you've been lost in a daydream a good portion of your life, and that the daydream is in many ways what you used to give the most significance.
But thoughts are not the enemy. Your own attachment to the thoughts is what prevents peace. And as you do your regular meditation, you naturally and easily feel less and less attached to the thoughts, and you become more and more happy and peaceful.
Adam Khan is the author of Principles For Personal Growth, Slotralogy, Antivirus For Your Mind, and co-author with Klassy Evans of How to Change the Way You Look at Things (in Plain English). Follow his podcast, The Adam Bomb.