Mindfulness has mostly been divorced from its actual context in Buddhism, so it's no wonder you're confused. In Buddhism, the cheif persuit is learning to break what might be called "the illusion of ownership." Mindfulness isn't an end, but rather a means to an end - the idea is to observe what's happening in relation to the feeling of beingsomething, or the feeling of control over things, and learning to see that it isn't actually you doing anything.
For example, when walking mindfully, you might observe that the steps happen on their own accord, as does your breathing, the thoughts that come up, your reaction to each thought, the emotions you feel - on and on. Eventually you start to realize that every aspect of your life is driven by cause and effect (or karma) and there is no separate central controller that is making decisions or doing actions independent of a cause.
Mindfulness of breathing has the added benefit of training concentration. The reason to practice concentration is to allow yourself to be more aware of this process of cause and effect happening, and give you the ability to make changed to your system of reactions. Each time you remove your attentionfrom a distraction and placing it back on the body and the breathing, you're exercising the ability to control the scope of what exists in your consciusness. By narrowing this down to one, or just a few objects, it gives you less things to identify with. Once you have disidentified with everything that currently exists in yor consciousness, you have removed all internal obstacles. It's best explained as a complete lack of cognitive dissonance - or perfect contentment with everything as it is and as it's unfolding - pure effortlessness.
Eventually, the goal is to stop the need to narrow the field of objects that come into your consciousness in order to let go completely. When you're achieved this, you are considered an "Arahant" - which is someone who has attained Nirvana, or complete unbinding. Buddhism views each person as a tangle of impersonal influences. The final goal is to completely untagle this set of influences and realize it's "empty" - there's nothing extra at the center that you should feel the need to say, "this is mine, this belongs to me."
The Buddha uses an analogy for this. He says, if you consider a cart, it's made up of wood, nails, an axel, wheels, etc. How much of this would you have to remove from a cart for it to stop being a cart? The line between cart/non-cart is arbitrary. The cart is made of trees, and metal rocks, and pitch made from long dead animals. When the Buddha looks at the cart, he sees both a cart (the conventinal, arbitrary label we use to define the object) and emptiness (a long, endless chain of cause and effect going back into unknowable history). The same can be applied to people. You look at yourself and define certain boundaries and say, "this is me, and this is not me." But suffering arises when the things you think of as you fall out from under your control.
Maybe you say, "The body is me." The Buddha would counter with, "If it's you, it should be under your control, but I could cut off your arm. Would that make you less you?" You might concede the point and say, "Maybe not my body, but then my feelings and mind are me." He might say, "I could insult you and make you angry or sad, if these feelings are you, why don't you control them?" So maybe you concede that feelings don't really belong to you, but certainly your thoughts and awareness do! But even this, when you observe it, seems to be divorced from a central, independent controller. Your thoughts arise in response to stimulus or in a chain from other thoughts. Your awareness goes towards things as it's attracted to them and moves away from things as it's repelled from them. Here the Buddha says, "If you don't control these things - nevermind whether they are you or not - do you think they're worthy of holding on to?"
So the Budha says the correct way to view the world is that it has no actual objects, no selves, no particulars. Everything is interdependent and connected to other things. Drawing lines over reality is only a useful convention - but we are completely convinced that this reality made of objects is real. When you insult me, I see you as attackingme - a visceral object that I am and identify with - but actually what's happening is you're pointing out an object that I aquired through cause and effect. Maybe you say I'm ugly, but you're insulting this body, not me. The body was made by nature and DNA - I had no say in the process and, thus, no real reason to be insulted. By trying to hold on to a specific set of these things and control them, we create suffering for ourself. So the key to lasting contentment is to let go of ownership of as much as you can.
By paying attention to what's happening, you can peer into the tangle of assumptions your mind is making and question them. Am I the one walking? Am I the one thinking? Am I the one paying attention? Eventually, when you see that you aren't, your mind lets go of "clinging" to that object, and it can function smoothly and effortlessly on its own.