Spirituality is a word that encompasses more than any one particular concept.
I believe spirituality is closely tied to meditation and the ability to be in the present. To be able to achieve a higher state of spirituality it is essential to live in the present. Below I will outline why breathing and meditation can lead to higher state of spirituality.
Breathing Meditation & Spirituality
There are many meditation paths, among them, breathing is by far my favorite. You can do this while standing, walking, sitting or even waiting in line.
Listen to the noise around you. Accept that there is sound, and try to hear as much of it as possible. By doing so, it is easier to block it out, since you have isolated that which you are blocking.
Begin breathing meditation. Breathe in deep and breathe out. Do this slowly, again and again. As you do so, think of nothing but the air as it enters your system, and leaves your system. In doing so, you might start slowly feeling how the breathing is affecting your body.
Once you are in a breathing rhythm, try to shut down your mind even further. Do not think of anything past or future. Try to actively just concentrate on an infinitesimally small amount of time which is now. When you first do this, you might still be occupied by immediate pasts, such as ten seconds ago. With time, you will be able to cut that time down significantly.
The point of this exercise is to try and achieve a state of "nowness". Once your immediate attention is very close to the present, you will feel a certain state of higher spirituality.
Final Notes
This might not work the same for everybody. This is something that works really well for me. It is also something that gets significantly easier to perform with practice. Good luck.