There are several ways one might attempt to answer this question, and I will try to enumerate them:
1) Jesus was so charismatic, and his hold over his followers so great, that their cultural Jewish inhibitions were overcome.
2) Jesus never said any such thing. Rather, the whole legend of the Last Supper was created by later Christians. There are two ways this could have happened:
2a) By influence from pagan religions, such as the cult of Dionysus, that already celebrated the eating of flesh and drinking of wine (the blood of the god),
2b) By a natural development within Christianity, uninfluenced by pagan religion.
Most New Testament scholars accept Paul's account of the Last Supper (in 1 Corinthians 11:23-34) as based on a historical event, and therefore assert some version of (1). Paul is thought to have written his letters about 20 years after Jesus's crucifixion, but to have been converted to Jesusism (the term "Christianity" did not yet exist) only a few years after the crucifixion. The early date of his conversion, and the direct report of the Last Supper "event," are enough to establish the authenticity of the event. Or so it is thought.
There are some serious problems with this conclusion, though.
First, I don't think these scholars really consider what it would have meant for a first-century Jew to drink "blood" - even if it was clearly symbolic blood. Imagine Oprah, on her last show, taking out a large Tootsie Roll and passing out pieces of it, saying, "This is my shit. Take it and eat it in memory of me." How many people would leave feeling good about Oprah? How many would actually adopt this ritual? (OK, probably a few nuts would. But Oprah has millions of admirers, Jesus only had 12.)
The gospel of John reports a very different Last Supper, one with no mention of Jesus's body and blood, and no command to repeat the event in remembrance. This is not because John (the author of the gospel, who certainly wasn't the apostle) rejects the body-blood deal: John 6:56 says
Whoever feeds on my flesh and drinks my blood abides in me, and I in him.
So it seems John accepts the Eucharist and its standard interpretation, but doesn't connect it with a Last Supper event.
The Didache, a first or early 2nd-century document, gives not one but two versions of the Eucharist (in 9 and 10), neither of which have any mention of the body and blood, or any connection of the ritual with a historical Last Supper.
Also, there is a well-known tradition of agape meals among early Christians. Like the Eucharist, these were shared meals, but there was no body-blood liturgy associated with them.
So, which is more likely: that Jesus instituted the Eucharist and commanded his followers to perform an abomination and to repeat it in his memory, and then some groups forgot that he had done so (but continued to celebrate some version of it anyway), or that the ritual as we know it was a later invention?
John Dominic Crossan argues forcefully for the latter option in The Historical Jesus (360-367). But he has a problem explaining how the blood-and-guts Last Supper appears so early in Paul.
Now, in my essays on early Christianity, I tried to stay with arguments and conclusions that are fairly widely accepted among New Testament scholars. What follows breaks this tradition, so, reader, beware!
What if the Last Supper in 1 Corinthians sounds like a later version because it is a later version - it was never there in Paul's original letter. That is, 1 Corinthians 11:23-34 , or some portion of it, is what scholars call an interpolation - a segment that was added to the letter at a later time. Many New Testament books are known to have interpolations. In some cases, one version of the text contains a note in the margin, while a later version contains the same note incorporated into the text. One well-known interpolation that is directly relevant is Luke's version of the Last Supper (Luke 22:14-23), which exists in a long version and a short version in different manuscripts. Thus, either someone added to Luke's original Last Supper story, or someone deleted something from it. (In the latter case it would not, of course, be an interpolation.)
Why might the 1 Corinthians passage be an interpolation? The main indicators scholars use (when there isn't direct evidence from textual variants) are vocabulary, style, and context. Now, the vocabulary here is solidly Pauline - there are no words that appear out of place (by reason of not appearing in other of Paul's letters). But this passage is the only place in Paul's letters where he tells a story from Jesus's life, and it is the only place where he has a direct quotation of Jesus's words. (There are about a half-dozen places where Paul gives an indirect quotation.) So the argument from style seems pretty strong. The context of the passage has a different understanding of "body" in mind: the body of believers in Jesus. Also, Paul has already given his version of the Eucharistic tradition in 1 Corinthians 10:16, so why would he repeat it here?
As I said, most scholars don't accept the argument for an interpolation here. (R.D. Richardson is the exception, from whom most of the above argument comes. See his commentary to Mass and the Lord's Supper, pp. 598-600.) So take the previous paragraph with as much salt as you like.
Still, the evidence points to the conclusion that Jesus did not "institute" the Eucharist, with body and blood symbolism, in a historical Last Supper. Rather (Crossan argues), Jesus established a tradition of eating shared meals at which all social ranks were able to eat together. This tradition got developed in various ways: as an agape meal with no symbolic overtones, or with the bread representing the body of believers (as in the Didache), or with the bread representing the body of Jesus (as in Mark, 1 Corinthians, etc). But where did this last interpretation come from? I'll try to address (2 a & b) next time.