You might be familiar with the usage of "as + [adj/adv] + as" to compare things. Sometimes it is used to emphasise something.
Sentence: The ant was as big as a dog.
Meaning: The ant was really big!
It is an objective truth that the dogs are bigger than ants. A concrete, visible comparison is being made. Everyone knows the degree to which this is true (that is, everyone knows that dogs are a lot bigger than ants).
If you want to say an ant is really big, instead of saying "really big" you can compare it to something that is a lot bigger than it.
Sentence: As angry as I was when he left us, building this rocket with him made me an engineer.
Meaning: I was very angry when he left us, but I appreciate the fact that I got to spend time with him before he left us, and we spent that time building this rocket, and building this rocket made me an engineer.
Here, there is no objective truth or concrete comparison that can be made. We do not know the degree of the speaker's anger. Only the speaker knows how angry he was, but it is implied that he was very angry.
In both cases "as" is used as an adverb to talk about the degree of something.
How big was the ant? Very big.
How angry was the person? Very angry.
You are right that "even though I was angry" can be used to replace it, but that removes the degree of his anger. So, I would say: "Even though I was very angry when he left us, building this rocket with him made me an engineer."
As hard as it was to learn to drive, I knew I had to do it. = It was very hard to learn to drive, but I knew I had to do it. = Even though it was very hard to learn to drive, I knew I had to move on.
As brave as Jake tried to act, he was still just a scared child. = Jake tried a lot to act brave, but he was still just a scared child. = Even though Jake tried to act very brave, he was still just a scared child.
To summarise: this construction is an adverbial phrase that emphasises something.