The subtle body is the emotional and thought body. Our emotions are connected to the chakra system, the energetic body existing with our physical body. The subtle body affects our yoga practice by constantly pulling us out of the moment and into our mind chatter and suppressed emotions.
In yogic teachings, each person has five bodies or koshas, that exist in a different concentration of energy. The physical, vital force, mind, wisdom, and bliss body. These five bodies are a reflection of the chakras.
Have you noticed how your mind begins to work over time as soon as you sit down to meditate? This is your subtle body trying to bring your attention to the things you are not wanting to look at. The emotions that you have buried somewhere in your psyche. It could also be all kinds of thoughts that flood in as you do your asana practice.
(Learn more in Find the Mind's 'Off Switch': 3 Ways to Rein In Your Brain During Meditation.)
The way around this distraction is to do one's best to become an observer of the emotions and the thoughts, whether they are negative or positive. Try not to run away from them, but rather to embrace them.
Try your best not to avoid feeling the emotions that come up. They must be felt to be healed. Suppressing them will not make them go away. Treat them like a baby and locate the emotions within the body.
(Learn more in Managing Your Emotions: The How and the Why.)
The term subtle body is a bit misleading as we tend to think of subtle as an esoteric concept. The subtle body is as real as the physical body. After all the entire universe is made of energy manifesting in different forms.