what is the "trial-averaged neural response function"?
ρ(t) in Equation 1.1 cannot be summed and averaged because the idealized spikes are infinitesimally narrow and the chance of two spikes from different trials occurring at the same time t is zero.

In Equation 1.2, the idea of convolving each idealized spike with a "well behaved function" h(t). If you use a rectangular function around zero as h(t), with a reasonable width (say 1ms), then Equation 1.2 will give you a timeseries that can be averaged. This is because the chances of two nonzero values from different trials occurring at the same t will become a more reasonable (nonzero) value. 

If you have enough trials, then your trial-averaged neural response function using this more reasonable representation of spikes will give you a good sense of the average spike density.