I disagree with the other answer. First, the structure here can be unpacked to the following (rather repetitive) equivalent phrasing:
She told me all her pitiful worries and woes:
- she told me how her brothers tease her;
- she told me how she had to come to school with secondhand robes and books;
- she told me how she didn't think famous, good, great Harry Potter would ever like her...
Now to understand this correctly, you need to understand that tell how is a bit of a set phrase here, and so "how" does not have its usual meaning. (Note it is not a phrasal verb like "tell off".)
Basically, in a situation where one person is recounting some information to another, "A tells B how X happened" means something like "A tells B that X happened, but maybe not in those exact words, and also maybe with more details provided".
For example, suppose a new student came to me (the teacher) and said "I don't like this school, I don't have any friends here." Later that day, when I meet with the principal I might say "I spoke with Mary, and she told me how she feels isolated."
Note that she did not say anything about the manner in which she feels isolated (which would be the usual meaning of "how"). Instead, this sentence basically means "she told me that she feels isolated". So why would I use "how" instead of "that"? The reason is that I would generally only say "she told me that she feels isolated" if she herself used the word "isolated". This is not a hard-and-fast rule, so you cannot always rely on it, but the use of "how" is a signal that I am paraphrasing what was said. You see this in the excerpt from Harry Potter: when Tom Riddle says "famous, good, great Harry Potter", this is a mocking paraphrase of Ginny's attitude toward Harry.
The use of "how" also signals that one is summarizing the recounted information. If he had said "she told me that her brothers tease her, that she had come to school with secondhand robes and books", and so on, it would suggest that she only said those three sentences, which would seem rather abrupt of her. Again this is not a hard-and-fast rule, just a connotation.
By the way, when "told me how" is used in this way, it can generally be replaced by "told me about how" without changing the meaning.