How to interpret neuron spiking models one comes across in literature?
Read dayan and abbot "theoretical neuroscience" 
Learn differential equations 
Know the relationship between voltage, current, resistance and conductance
Differential equations is absolutely essential though. you don't need to learn to solve them (the computer will do that for you), you just need to learn to know what they mean.    
How do researchers translate observations of nature into symbolic, mathematical approximations/models of these phenomena? By gaining intuition of how the model components relate to empirical observations. it is relatively rare that someone comes up with a new model from scratch. most of the time people retool or remix previous models.