Firstly, there are many ways to construct a suffix tree. There is the original O(n) method by Weiner (1973), the improved one by McCreight (1976), the most well-known by Ukkonen (1991/1992), and a number of further improvements, largely related to implementation and storage efficiency considerations. Most notable among those is perhaps the Efficient implementation of lazy suffix trees by Giegerich and Kurtz.
Moreover, since the direct construction of suffix arrays has become possible in O(n) time in 2003 (e.g. using the Skew algorithm, but there are others as well), and since there are well-studied methods for
suffix arrays are usually preferred over suffix trees. Therefore, if your intention is to build a highly optimised implementation for a specific purpose, you might want to look into studying suffix array construction algorithms.
However, if your interest is in suffix tree construction, and in particular the Ukkonen algorithm, I would like to suggest that you take a close look at the description in this SO post, which you mentioned already, and we try to improve that description together. It's certainly far from a perfectly intuitive explanation.
To answer the question about how to compare input string to edge labels: For efficiency reasons during construction and look-up, the initial character of every edge label is usually stored in the node. But the rest must be looked up in the main text string, just like you said, and indeed this can cause issues, in particular when the string is so large that it cannot readily be held in memory. That (plus the fact that, like any direct implementation of a tree, the suffix tree is a data structure that contains a lot of pointers, which consume much memory and make it hard to maintain locality of reference and to benefit from memory caching) is one of the main reasons why suffix trees are so much harder to handle than e.g. inverted indexes.
