Why sigmoid curves appear in quantitative analysis of many biological phenomena?
I would suggest following: take any one-dimensional process that is described by gaussian function. If you integrate such process along variable, you will get a sigmoidal curve. For details see normal and cumulative distribution.

Now, the question is: how many biological processes are described/modeled by Gaussian distribution and dependencies? A lot!

Let's take a look at dinosaur growth from Revisiting the Estimation of Dinosaur Growth Rates



This figure describes how size of dinosaur changes with some time (model, I assume). Remember link between gaussian and sigmoidal functions? If you differentiate, that is find how fast aforementioned sigmoidal function changes, you will get gaussian bell-curve! Which tells you: dinosaur grows faster during middle age, and slower before and after that.

DNA denaturation curve is sigmoidal because efficiency of denaturation is again gaussian. At low temperatures very limited amount of molecules will denature, but at high temperature there are no more molecules to denaturate left, so the bell-curve. Anothe explanation goes like that: at low temperature additional heat doesn't change much because it is still too "cold", at higher temperature same amount of heat is just a drop in a bucket, barely noticeable.