That rare and wonderful fire in the chest feeling can, as you were wondering, be explained biologically. Any time we experience any emotion, positive or negative, neurotransmitters (hormones, or chemical messengers) are released by certain parts of the brain into the central nervous system, which communicates with every cell in the body.
Rage, love, (enthusiasm,) awe, disgust, delight; in fact every emotion of which we are capable, and our reaction to any situation we may encounter, causes the release of a variety of chemicals specific to the stimuli experienced. Our previous experience, as well as instinctive reactions, contribute to what the brain decides to release.
A perfect example everyone has felt is what happens to your body when you suddenly feel fear, or anger. The brain sends instant messages to your adrenal glands, which immediately release stress hormones, cortisol and adrenaline, into your bloodstream. This causes your heart to beat faster, your veins to constrict, your lungs to breathe faster and more shallowly, your senses and perceptions to become more acute, etc, to prepare you for action. Any form of stress produces some stress hormones.
On the other hand, when something makes you feel great, that is because the brain has released endorphins, hormones which affect us profoundly. Love can cause a veritable cascade of them, especially when "falling in", and some are released whenever something gives you pleasure. When that indescribable sensation of a fire within us, or that we are going to explode with something or other, it gets quite complex, because there are so many things going on mentally, and as a consequence, physically. (Actually, nothing we experience is either 'mental' or 'physical' ~ they are inseparable ~ so it is meaningless to try to classify them as two different systems)
Anyway, with what we are talking about, there is a powerful feeling of excitement, sometimes a rush of - hmm -happiness may not quite describe it, but close, maybe? - glee? joy? What-evverrrr... a feeling of being thrilled by something... add your own here. For this, the brain can produce several concoctions of both stress hormones and the "happy" ones, (such as serotonin and dopamine) and this combination can really excite the cells within many of the areas affected, especially your stomach, heart, lungs, skin, (even intestines, at times) creating that wonderful feeling of a fire burning within.
Personally, I think that knowing what actually happens within us is far more incredible than the less scientific explanations.
