Learn English – Cross Origins of Comrade and Camaraderie

Etymonline.com says the meaning of making somebody self-conscious is first recorded in 1828 and shows a French and Italian origin:

1670s, "perplex, throw into doubt," from Fr. embarrasser (16c.), lit. "to block," from embarras "obstacle," from It. imbarrazzo, from imbarrare "to bar," from in- "into, upon" (see in- (2)) + V.L. **barra* "bar." Meaning "hamper, hinder" is from 1680s. Meaning "make (someone) feel awkward" first recorded 1828. Original sense preserved in embarras de richesse (1751), from French (1726): the condition of having more wealth than one knows what to do with.

The OED says the etymology is from "French embarrasser, lit. ‘to block, obstruct’, < embarras" and their first quotation for making someone feel awkward is from Webster's 1828 An American dictionary of the English language.

Here's Webster from 1828 (plain text):

Q: and how did it come to have the meaning of making somebody self-conscious and abashed?

It originally meant to literally encumber, to hamper or to impede; or to perplex, to confuse. It later then came to the modern meaning, as in OED's definition 2.b.:

To make (a person) feel awkward or ashamed, esp. by one's speech or actions; to cause (someone) embarrassment.

So one's speech or actions are stopping and perplexing the other, which causes the feeling of awkwardness.

Here's some examples from the introduction to Webster's 1828 dictionary that show the different meanings:

• I found myself embarrassed, at every step, for want of a knowledge of the origin of words

• Similar contractions have taken place in all other languages; a circum- stance that embarrasses the philologist and lexicographer at every step of his researches; and which has led to innumerable mistakes in Etymology.

• Lexicographers are often embarrassed to account for the different signifi- cation of words that are evidently derived from the same root.

• This practice of blending with the English many words of an orthography, which in our language is anomalous, is very embarrassing to readers who know only their vernacular tongue, and often introduces an odious difference between the pronunciation of different classes of people

• On the other hand, all that I have seen, serve only to obscure and embarrass the subject, by substituting new arrangements and new terms, which are as incorrect as the old ones, and less intelligible.

Merriam-Webster Online gathers several related senses of take:

11 b (1) : to obtain as the result of a special procedure : ascertain <take the temperature> <take a census> (2) : to get in or as if in writing <take notes> <take an inventory> (3) : to get by drawing or painting or by photography <take a snapshot> (4) : to get by transference from one surface to another <take a proof> <take fingerprints>

The common meaning here is to record something by procedure or writing instrument.

I hadn't previously seen the uses take a painting or take a drawing, so I consulted Google books and found that taking a drawing was common in the 1800s, during the rise of photography. This sense of take appears to have arisen in the late 1700s; note this example from The New-York Magazine or, Literary Repository (1792):

Mr. Peale, we hear, is engaged to take a painting of this extraordinary person, to preserve to future time the features and form of a person furnished with nerves and constitution to exist to so surprising an age, on that ocean of time which has long ago swallowed up so many millions of his contemporaries.

While making a drawing was always more common than taking one, I suspect that the latter usage took hold for photographs because the process of capturing (taking) a photograph is distinct from printing (making) it, and the taking happens when you open the shutter.