Why is "can" such an odd verb?

The English verb can is very strange for several reasons:

• It drops the to on any infinitive verb forms that follow it. That is, unlike in the verb want in the sentence I want to eat, you would not include the infinitive to marker in the sentence I can eat.
• Can itself has no infinitive form. A construction like I like to can understand foreign languages is not grammatical; you have to use the phrase to be able, as in, I like to be able to understand foreign languages.
• Can does not conjugate. Most verbs add an s or es when they have a third person singular subject, but can does not.
• You cannot add tense modifiers to can. I will can, I canned, I had canned and so on are all ungrammatical (unless you are talking about putting things in can). You have to use constructions like "I will be able to", "I could", "I had been able to", etc.

All of the other languages I know anything about do not share any of these characteristics in their corresponding verbs.

What are the historical reasons for why the verb can behaves so oddly in English?

As Peter Shor indicates in his comment, these are modal verbs and are found in other Germanic languages as well. Their unusual conjugation is due to their historical source:

Germanic modal verbs are preterite-present verbs, which means that their present tense has the form of a vocalic preterite. This is the source of the vowel alternation between singular and plural in German and Dutch. Because of their preterite origins, modal verbs also lack the suffix (-s in modern English, -t in German and Dutch) that would normally mark the third person singular form.

Preterite-present verbs are a very old group of verbs found in Germanic languages that in turn descended from a set of Proto-Indo-European verbs. Originally these were perfect-aspect verbs that referred to a present state.

The common characteristic is that the "present tense" of these verbs was conjugated like the "past tense" (essentially). In Modern English this now means that the present-tense "-s" ending is not used, and that modifiers like "to", "will", etc. can't be added. It happens that these verbs are all modals in Modern English, which probably helps preserve the unusual conjugation.

So this is a bit of a historical accident that made a lot more sense in PIE, and now is frozen that way.

Modal auxiliary verbs in English are all irregular, idiomatic, defective (which means they are missing normal inflected forms, viz. infinitive, past tense, past participle, and 3rdSg present), and very, very common. All the phenomena mentioned in the question are typical of modals, so can is simply a normal modal in this regard.

Can does have a couple of unique peculiarities, though. All modals have at least a logical Epistemic sense, about possibility and probability:

• This can't be the place.
• This may be the place.
• This must be the place.
• He should be asleep by now.

and a social/personal Deontic sense, about permission and obligation:

• You can't carry that on the plane.
• You may leave your bag here.
• She must leave immediately.
• You should see a doctor about that.

and can is no exception.

However, Epistemic can occurs only in NPI environments, and is usually ungrammatical without a Negative trigger.

• *This can be the place.
• *She can have meant that.

and can also has an additional sense, referring specifically to personal ability, rather than to permission or logical possibility:

• She can run a marathon in under 2 hours.