I know たい means "want to." But the examples below confused me. Especially the
second sentence translates to "I want to eat." I don't know why this is the
translation. I'm not sure what is the subject in the second sentence. If I use
"I", then it would be awkward because it would mean "I am want-to-eatable"? Is
it a weird exception? If the actual subject is "I", I'm not sure why would たい
mean "want to eat" instead of "want-to-eatable."

僕は魚が食べたい。

lit: As for me, fish is want-to-eatable.

僕は食べたい。

lit: As for me, [SUBJECT] is want-to-eatable.
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question
grammar syntax particle-が subjects
Subject confusion in ~たい sentences
In a clause, both the subject and the object for 食べたい are marked with が. In
other words, "(that) I want to eat fish" is expressed as 私が 魚が 食べたい（こと）.

In a sentence, however, you have to add は to either or both of the subject and
the object. When you add it to the subject, the sentence becomes 私は 魚が 食べたい.
Likewise, 魚は 私が 食べたい for object, and 私は 魚は 食べたい for both.
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There's a lot of confusion about Japanese grammar, that is often caused by
trying to understand the Japanese grammar through the lens of the English
translation. English and Japanese are structurally _very_ different, so it's
not at all surprising that this approach would muddle things.

  * As an example, you start out by saying, _“I know たい means "want to."”_  
  
Analyzed differently, all ～たい forms are adjectives that have no objects (since
adjectives can't have objects), and these adjectives describe subjects as
"desirable" for having the action of the verb performed upon them.

  * You also say, _“I'm not sure what is the subject in the second sentence. If I use "I", then it would be awkward because it would mean "I am want-to-eatable"?”_  
  
When using は (in simpler sentences like these), you're not marking either
subject or object -- you're marking the topic, the outermost context about
which the rest of the sentence describes the circumstances.

Let's look at your sample sentences.

## Sentence 1: 僕【ぼく】は魚【さかな】が食【た】べたい。

As you note, a direct translation could be, _“As for me, fish is want-to-
eatable.”_ We could render this slightly differently, as _“[[As for]{は}]
[[me]{僕}], [[fish]{魚}] is **[[desirable for]{～たい}] [[eating]{食べ～}]**.”_

An analysis:

  * 僕【ぼく】は  
The 僕【ぼく】 is marked as the topic with は, so we know that the 僕【ぼく】 is the
outermost context, about which we are then talking about other things in
relation to that outermost context. English doesn't have this distinction in
the same way, so we make do by using the slightly-awkward phrasing, _" as for
~,"_.

  * 魚【さかな】が  
The 魚【さかな】 is marked as the subject with が, so we know that the 魚【さかな】 is the
inner context, about which the following verbs or adjectives would apply.

  * 食【た】べたい  
While this is _translated_ as a verb phrase, _" want to eat"_, using the two
verbs _" want"_ and _" eat"_, this 食【た】べたい is technically an adjective that
describes some **quality** of a thing. In this sentence, it's describing the
subject 魚【さかな】 as having the **quality** of _" desirable for eating"_.

## Sentence 2: 僕【ぼく】は食【た】べたい。

As you note, a direct translation could be, _“As for me, [SUBJECT] is want-to-
eatable.”_ Tweaked as above, we might have, _“[[As for]{は}]
[[me]{僕}],`[SUBJECT]` is **[[desirable for]{～たい}] [[eating]{食べ～}]**.”_

An analysis:

  * 僕【ぼく】は  
The 僕【ぼく】 is marked as the topic with は, so we know that the 僕【ぼく】 is the
outermost context, about which we are then talking about other things in
relation to that outermost context. English doesn't have this distinction in
the same way, so we make do by using the slightly-awkward phrasing, _" as for
~,"_.

  * 食【た】べたい  
While this is _translated_ as a verb phrase, _" want to eat"_, using the two
verbs _" want"_ and _" eat"_, this 食【た】べたい is technically an adjective that
describes some **quality** of a thing. In this sentence, ...  
  
We don't have a subject!  
  
Oh no!

... except, Japanese is a language that allows for lots of omission. Consider
these simple transitive sentences:

  * 僕【ぼく】は魚【さかな】を食【た】べる。 
    * → We've got a topic, and an object. Considering the context of that topic, we also know that the subject -- the agent of the verb, the person or thing doing the action -- is the same as the topic.
  * 魚【さかな】を食【た】べる。 
    * → We don't have a subject!
  * 僕【ぼく】は食【た】べる。 
    * → Hey, wait a minute -- we don't have an object!
  * すぐ食【た】べる。 
    * → Ack! Now we don't have _either_ a subject _or_ an object!

Despite the challenges of not having various pieces of information, Japanese
grammatically allows all of the above, and listeners (or readers) of the above
will still understand what is being talked about.

Getting back to your sample sentence #2:

  * 僕【ぼく】は食【た】べたい。

Unless this is a very strange context about monsters, cannibals, or the like ,
we can be reasonably sure that the 僕【ぼく】 is not the subject of the adjective
食【た】べたい -- this is not saying that the _“[I]{僕}”_ is _“[[desirable for]{～たい}]
[[eating]{食べ～}].”_

Instead, what we have here is an e̲x̲plicit topic, marked with は, and an
i̲m̲plicit subject, which is simply omitted. This is a parallel kind of
construction as 僕【ぼく】は食【た】べる, where we don't explicitly mention what is being
eaten. In 僕【ぼく】は食【た】べたい, we don't explicitly mention what is desirable for
eating. We know that _something_ is desirable for eating, but we don't have
the specifics for what it is -- this is saying that, _“[[As for]{は}]
[[me]{僕}],`[something unspecified]` is **[[desirable for]{～たい}]
[[eating]{食べ～}]**.”_

By extension, we know that the desire to eat is what we're talking about, with
regard to the circumstances of the outermost context, 僕【ぼく】.

So in translation, we wind up with something that is grammatically quite
different from the source Japanese, simply because English is grammatically
quite different. Instead of saying, _“[[As for]{は}] [[me]{僕}],`[something
unspecified]` is **[[desirable for]{～たい}] [[eating]{食べ～}]** ”_, we say
_“[[I]{僕}] **[[want to]{～たい}] [[eat]{食べ～}]**.”_

* * *

Please comment if the above does not address your question, or if you find it
confusing, and I can update the post accordingly.
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answer
Let's see if I can clear some confusion. In Japanese as opposed to English は
does not function the same way 'is' does in English. It is a topic marker
marking what the sentence is about. In the case of 私は食べたい。 The topic is 'I'
and the verb is "want to eat'. It would be kind of like saying, "As for me, I
want to eat." The closest word to is, are, and other be verbs is actually である
and it is really formal Japanese. In conversation です is used in polite
conversation, though it is often dropped entirely for adjectives and the like.

One last thing I should mention is that, unless you are comparing your
preference to eat to someone else in juxtaposition to their own preference you
probably wouldn't use this structure too much. If the topic of 'you' is
already established, it is simply better to declare using が what exactly you
want to eat. Thus 私は魚が食べたいです。 can be shortened to 魚が食べたいです。. Hope this helps a
little.
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answer