Maybe I'm wrong, but I believe that when I'm sitting to improve myself, I have to watch my mind to see what is passing through it, and to figure out the reason for that thought and then let it go sometime. But when I'm sitting, my mind doesn't seem to work the same way as when I am not sitting. I'm just there, sitting and watching my respiration without thoughts. And when I get up and go do other things, all of the thoughts and feelings come back. What am I doing wrong?
You are not necessarily doing anything wrong. It depends on what kind of meditation you are practicing and what your purpose with the meditation is.
It sounds like you are practicing Samatha Meditation (tranquility meditation) but instead you would like to practice Vipassana Meditation (insight meditation).
In samatha meditation one keeps attention on a single object, e.g. the breath trying to make that object stable. This type of meditation is creating a unification of mind. The goal here is to build deep and powerful states of concentration. When the mind is concentrated, i.e. by only paying attention to one object, other objects disappear. They slowly stop arising. This is only temporarily. This only happens as long as the power of concentration is present. When concentration weakens or when one stops the session and goes up and do other things then slowly the unification wears off and other objects e.g. thoughts and ideas come back into the mind.
When doing samatha meditation one temporarily suspends mental activity. When finishing the session this suspension of mental activity can last for a while afterwards but will eventually stop.
In vipassana meditation one does not keep attention on a single object instead one is giving attention to whatever arises whenever it arises. One has the abdomen as an anchor (mahasi sayadaw tradition) which one comes back to after having observed or noted objects as they arise and fall in the present moment. The anchor is a tool that one uses so that the mind will not get lost and follow after objects.
One is keeping attention on the rising and falling movements of the abdomen. When a thought arises then one shifts attention and watches the thought. One notes it a couple of times saying "thinking, thinking" and then returns to the abdomen. If a feeling or a sound arises then one shifts attention towards these objects and notes "feeling, feeling" or "hearing, hearing". One can also note the feeling more specifically, e.g. if anger arises one notes "angry, angry" or "sad, sad" if sadness arises.
When doing the noting it's important to sent the mind out to the object and not to keep attention on the "mental label". So when noting e.g. pain in the leg one is noting "pain, pain" but ones attention is on the pain in the knee and not on the mental label.
In vipassana meditation one tries to be with the objects as they arise when they arise. The goal is to learn about their true nature, i.e. the 3 signs of existence (impermanence, unsatisfactoriness and the uncontrollable nature). One does not follow after objects. One also does not try to find out where e.g. a thought came from. This is not important. What is important is to see that thought as an object that arise and cease. One does not give special attention to any object at all.
Hope this helps. If you have any questions to anything i wrote let me know.