I think maybe you are anxious about thinking. 

Anxiety makes everyone dumber by activating the flight-or-fight sympathetic nervous system, which suppresses higher cognitive functions. And anxiety can make people dumb in a second way, by making them indecisive. That's because the down-sides of any course of action seem more severe. That includes accepting a new belief as a result of thinking.  So that could be most of what's going on.

I also wonder if your specific anxiety has attached to thinking, creating an ugh field around it.

These theories are a result of studying dopamine function in decision-making as a career, and having a long-term partner with clinical anxiety. 

I think clinicians would agree that anxiety makes people dumber in proportion to how anxious they are, that anxiety causes indecision, and that anxiety can attach to any specific topic, including "thinking". 

Indecision extending to not accepting beliefs as the result of thinking is my own theory, although I think it's likely to be correct. I think most clinicians would neither agree nor disagree on that, since it's more of a detailed neuroscience/cognitive psychology theory, and only a few specialists even think about higher cognition in that kind of mechanistic detail.

If some of that is correct, you might improve your thinking by being easier on yourself. Saying "yay me, I'm thinking about something even if it's not going as well as I'd like" might be a good step. Focusing on goals appears to be the best known approach to anxiety; see the Acceptance and Commitment Therapy overview, which appears to have better results than any other clinical approach to anxiety.

Finally, let me point out that you can in fact think rather well in some cases, because this question contains a well-thought-out theory of thinking, and applies it to you in a unique way. So I'm pretty sure you did some good thinking to come up with this detailed and unique question. However you managed that might be a road map to thinking better.

[disclaimer: mostly responding to the title]

Anxiety implies Sympathetic Nervous System activation. SNS is [simplistic model incoming] great at sensory awareness and physical movement, but bad at nuanced thinking and incorporating new information. I got better at calming my SNS using Somatic Experiencing Therapy, especially emotional titration. 

I expect that a lot of whether or not you "dare to think" is about the social feedback you get for thinking and voicing ideas. 

If you are in a social environment where you get punished for voicing your own ideas, that might lead to a lot of anxiety about them and not voicing your own ideas. If you are in a social environment with a bunch of rationalists that expect you to have your own ideas I would expect that it would come much easier to you to voice them.

I have two points/notes:

1. Your post here made some self-contradictory-seeming claims. I'm not criticizing, just suggesting that you may benefit from additional work to clarify and articulate a coherent account of what you're struggling with. For example, the title of your post is "Why does anxiety (?) make me dumb," but that overemphasizes anxiety as a hypothesis relative to the rest of your post. You also claim both that "I’m stuck at... ‘understand the content of the statement’" and "I have no trouble understanding reasonably complex concepts, but just never will discuss them, think about them, or even apply them," and characterize yourself as a "human encyclopedia." Often, articulating the problem accurately means you're 90% of the way to solving it.
2. I have personally experienced that a multi-year project of deliberately trying to learn how to learn has yielded massive improvements in my ability to both learn and think. There were a huge number of mistakes along the way, but it all has culminated into a level of skill that I'm really proud of, and has come despite the fact of me aging and the clear loss of certain mental faculties I possessed at a younger age. I think I am worse at automagically learning the way I used to, but better at learning overall via deliberate conscious effort. Yet my effort in this area depended a lot on the ability to use self-reflection and rational thinking to introspect, propose hypotheses, try things out, iterate and improve, and if you're feeling bottlenecked in that area you might need to work on that. 

A random stupid thought that occurs to me is that maybe your limbic system might be set to be too trusting of the truths you have "already accepted", and then maybe something else in your limbic system has been hurt enough to feel like "actions based on beliefs get me hurt" and so it has shut down that whole category of "theoretically motivated actions"?

Naively, two such mechanisms hiding in your limbic system would, together, perhaps create the totality of behavior and mindset that you describe?

There is a sequence of posts on babbling and pruning that is not on LW itself, but seem "worthy of the canon" to me, that you might not have read. (Oh. Apparently they got on LW and here's the fifth one!)

If we combine the babble & prune frame, plus the hypothesis about what might be in your limbic system, it suggests that maybe your prune submodule  got overactive and your babble submodule noticed that it was being totally ignored and lost its enthusiasm... or something?

I could imagine numerous exercises that might change either module on purpose, like "doing some improv where you only say yes and everyone love bombs you for being so good at trying new things" or maybe like "writing on the same prompt, over and over, trying on purpose to be hilariously wronger and worse, but also different, with each rewrite attempt" or maybe like comfort zone expansion?

Separately, with much much much less support from the local canon, I tend to be really big on safety engineering, and watching out for reversibility (which is a deep deep deep principal that motivates bayes, though sometimes in bayesian contexts "the same math" is referred to as a "consistency" criterion rather than a "reversibility" criterion).

The key larger point I'm trying to make "from outside local canon" is that maybe your pruning and babbling were basically correctly allocated less brain/time/effort resources!

Maybe (not necessarily, but just "maybe") you somehow weren't good enough (in your old contexts, at your past development levels) at either of those techniques to usefully expand skills in either of them in a safe way!

If you just go meta, and use "systematic willpower" to overrule what your limbic system may have done by default, then you might just stumble straight into the valley of bad rationality? Maybe!

So I would say to maybe "try babbling and pruning on paper about safely and reversibly doing improv". 

Then maybe try doing improv about hilarious unsafe and irreversible ways to do comfort zone expansion?

And then perhaps consult your parents on doing one tiny small real (potentially irreversible) comfort zone expansion expansion exercise?

(Or maybe skip all this nerd bullshit, and go get a job doing door-to-door sales for two months until your sales numbers are the highest, and then quit! Because maybe "the world is on fire and we should be speed-running our way to a win condition instead of tiptoeing around like a bunch of fucking nerds." (Synthesized speedrunning suggestion: do the writing exercise today, the improv exercise tomorrow, talk to your parents on day 3, and then find and say yes to a crappy temporary just-for-learning sales job on day 4.))

I've just been reading Anxiety Rx by Russell Kennedy (which applies to troublesome thoughts in general not just anxiety so think it is rather unfortunately titled!) and I think some of the ideas in it might be relevant here. 

His assertion is that it is easier to use the body to calm the mind than to use the mind to the calm the body. If you are a very cognitive person it is tempting to do the latter rather the former - you believe (falsely!) that you can talk yourself into feeling safer. 

He makes an interesting distinction between anxiety as the thoughts in the mind and alarm as the corresponding sensations in the body. You then have a looping feedback cycle between the two, where they can each make the other worse. 

However, the cycle is generally initiated by alarm (and he talks about two different types of alarm - foreground and background, with foreground being the universal fight-or-flight response and background being caused by trauma). Often alarm is initiated unconsciously, and then our brain  tries to come up with an explanation for that alarm. It sounds like it your case perhaps you have some unconscious trigger actually related in some weird meta way to thinking or the possible social consequences of thinking. 

His solution for breaking the feedback loop involves awareness of the sensations in the body, breathing and self-compassion, but I think other therapies that focus on the 'alarm' rather than cognitive strategies might just as valid. For me the ideas in the book fit very nicely with Internal Family Systems therapy and also with Acceptance and Commitment Therapy for example. Kennedy is clearly a big fan of Somatic Experiencing Therapy which I suspect has influenced his approach. 

>What would you say if I told you I’m stuck at the first step, the one that goes ‘understand the content of the statement’?

I would say proceed no further until you do understand, or accept that you will never understand, which will leave your mind free to focus on something else. Anxiety, for me, is defined as knowing something to be true that I refuse to believe. I have found it a helpful definition.