In simple words, how are body sensations linked to moods and feelings?

In simple words, emotion has three parts: the root, the stem, and the flower. The root is an imprint (samskara). The stem is a physical reaction. The flower is a psychosomatic sensation (vedana).

Emotion is a precursor of thought, it is like a very fuzzy/blurry thought. Before humans, animals already had some basic intelligence -- that's emotions. When a deer is grazing, her brain is watching for signs of danger. When she hears sudden noise, her brain creates a basic impulse to run away. The heart beats faster, the lungs breath faster, the muscles of the stomach contract, the legs start shaking. A basic formula like "sudden noise == danger" is an imprint or preconception -- the root of the emotion. The physical reaction is the stem of the emotion. And the way the physical reaction (heart, breathing, muscles) feels to you -- is vedana, the flower.

If you inspect your psychosomatic environment, you will surely notice subtle sensations in and around your chest, stomach, forehead, eyes, hands, wrists, feet etc. Subjectively, these feelings are often not located on the surface of, or inside the body, but rather "in the air" around it. Some of the emotions most obviously accompanied by such sensations are worry, urge, longing, resentment, irritation but also the positive ones e.g. love, fondness.

If we examine the root of the emotion more carefully, we can see that it works through evaluation (appraisal) by the mind of the present situation, against the basic assumptions and expectations (formulas, imprints, preconceptions) about how things are supposed to be. If the actual situation ("is") seems to be matching the expected situation ("should"), a positive emotion is generated (sukha), and if the two mismatch -- a negative emotion is generated (dukkha). By dropping the preconception, we prevent (nirodha) the emotion from arising.

For example, if you fall in love with a woman, and want to be with her, but she loves another man, and does not want to be with you -- you may feel depressed. If you look carefully at your somatic sensations, you may notice e.g. burning in your chest or dull pain in your belly etc. When you stare at this burning/pain long enough, you will eventually realize that it is caused by a mismatch between the mental picture of being with her, on one hand, and her not being available, on the other hand. This mismatch is called unsatisfied thirst, or tanha. If you keep looking, you will realize that the root of the emotion is your preconception that e.g. the woman has a perfect character on account of the way she moves, smiles, laughs, and flips her hair. (When you keep visualizing those attributes of her, and keep mentally connecting those with her - fictional - perfection, that is called "feeding the preconception", or upadana.) In order to stop suffering, you have to clearly see the preconception, and let go of it.

This is why Buddha taught kayagata-smrti (mindfulness rooted in the body) and anapana-smrti (mindfulness of breathing) -- because the root would be too hard to see under the ground, but the flower is highly visible. Once you notice the flower, you can start tracing it down to its roots (preconceptions) in meditation, and by extracting the roots (letting go of attachments, dropping identifications, dismissing the formulas, dissolving samskaras) you liberate from dukkha. Then you attain the state of suchness (tathata), when all your experience is basically sukha, because in suchness your "is" always matches your "should". When this practice matures, you go beyond dichotomy of right-and-wrong, beyond sukha-and-dukkha, and even beyond the beyond -- then you become completely rooted in tathata, and become what's called a tathagata.

Anyway, to get back to your question, emotions are always accompanied by sensations, however subtle. Somatic sensation is the most visible component of emotion. (The other component of emotion -- the mental obscuration or klesha -- is usually invisible, because it obscures the very ability to see oneself objectively.) So sensations, especially ones in the diaphragm area, but also in lower abdomen, throat, and forehead, are great windows into the "subconscious" mind.