Why are we transcribing prosody? and what is the philosophy behind it? In general what is the purpose of it?
It is way more efficient than the alternative. The alternative is to compute some acoustic measure over a chunk of speech in question, an analysis which takes a long time to set up and to perform. Auditory transcription on the other hand is remarkably quick, once the system has been set up. An implication behind any transcription is that there are a small number of discrete categories of the property, for example length might be reduceable to two or three distinctions (so that a plot of duration values would show two or three regions of high population, not uniform grey). There is also the presumption that the phonetic categories will turn out to correlate with some other cognitive state (generally, something pragmatic like sarcasm).
There are some counterindicating factors which indicate that the categoriality assumption is not completely true. This is where phonetic iconicity comes in. For example it is very common to find prolongation used on distal demonstratives (though it would be strange in English). If "there" is the ordinary word for a location away from you and me, prolongated "the-ere" would indicate "further away, "the-e-ere" would be "even further away", with the only limit on the number of distinctions being the non-linguistic ability to distinguish two durational patterns. Iconicity would be a kind of limiting noise on prosodic transcriptions, meaning that if an acoustic property signals "surprise" (which implies a categorial analysis "surprise is signalled with H"), there could still be continuous ranges of pitch raising signalling greater degrees of surprise. But you would not find for instance that "beyond 500 msc of added duration, prolongation signals shock rather than happiness".
We study prosody because we observe that it is a meaningful, functional component of language. We transcribe it with the goal of abstractly understanding the contrastive elements that speakers seem to be perceiving and producing (and the system of interaction for those elements). This is analogous to other aspects of linguistics; e.g., we could roughly say that (segmental) phonology seeks to understand the system of sound elements in human languages.