Q:

How does the prefix 'entre' in French's 'entreprendre' compare with the prefix 'under-' in English's 'undertake'?

enterprise (n.)
early 15c., "an undertaking," formerly also enterprize, from Old French enterprise "an undertaking," noun use of fem. past participle of entreprendre "UNDERtake, take in hand" (12c.), from entre- "between" (see entre-) + prendre "to take," contraction of prehendere (see prehensile). Abstract sense of "adventurous disposition, readiness to undertake challenges, spirit of daring" is from late 15c.

I heed the Etymological Fallacy, but how should the etymology be interpreted, to connect the prefixes 'entre-' with 'under-'? .
Update on 2015 Dec 4: I might have answered my own question. ELU references
pp 210-214, An Analytic Dictionary of English Etymology, An Introduction (2008) by Anatoly Liberman that explicate 'under-' in 'understand',
which is summarised here whose final 2 paragraphs recapitulate Etymonline below (which I failed to scrutinise):

[under (prep., adv.)] Also used in Old English as a preposition meaning "between, among," (though this may be an entirely separate root; see understand).
[...] the under is not the usual word meaning "beneath," but from Old English under, from PIE *nter- "between, among"
[...]
That is the suggestion in Barnhart, but other sources regard the "among, between, before, in the presence of" sense of Old English prefix and preposition under as other meanings of the same word. "Among" seems to be the sense in many Old English compounds [...]

A:

How did entre- "between" evolve to mean UNDER in UNDERtake?

It didn't. English used one prefix to derive this meaning from the more general root for taking; French used another. You can turn it around and ask, from a French speaker's perspective, how the English under "sous, sou-" evolved to mean "entre". While we're at it, Russian formed its word for undertaking by adding the equivalent of "fore-" or "pre-" to the equivalent of "receive" or "accept". The starting point may be more or less the same ("take"), the end point may also be the same ("undertake"), but individual languages simply took different paths to get from one to the other. I don't see a reason to use any specific version as a semantic standard.

A:

You are not wrong; German unter and Latin inter are actually related.  See for instance wiktionary PGm entry for under:

Etymology. From a merger of two originally distinct suffixes as a result of
  Verner's law: Proto-Indo-European *n̥tér ‎(“inside”), from which also
  Latin inter, and Proto-Indo-European *n̥dʰér ‎(“under”), from which
  Latin īnfrā, Sanskrit अध ‎(adha, “below”) and Ossetian дæлæ ‎(dælæ,
  “below”).

This explains why in OE, under had both the meaning of "beneath" and that of "amongst/between".  Only the former acceptation subsists today.
As an aside note regarding the latter meaning of "between", you can have a look at de Vaan 2008 inter (many cognates meaning "entrails" in many languages - gory).
The prefix under in undertake is to compare to unter in unternehmen ("undertake" in Present Day German) - since "nehmen" means "to take".
And as a matter of fact, before switching to "undertaken" in Middle English, Old English had undernimen.  But then, during Middle English, taken displaced OE niman - IMO under the influence of Old Norse. Undernimen was therefore logically replaced by undertaken.  Also note that, in its other meaning of "understand", undernimen also disappeared and only understandan remained as PDE understand.
French entre is, as you well know, a direct descendent of Latin inter.
Romance languages use inter or in 1. Italian has both imprendere/intrapendere - as you know prendere is to take.  Also Spanish has emprender; Portuguese has interprender.  
You have a similar phenomenon with underwritan ("underwrite") and unterschreiben).  But this time, German takes schreiben from Latin scribo and English sticks with  PGm *writana.  See subscribe.
Also worth mentioning is entertain (cognate with French entre-tenir etc); to compare with German unterhalten (halten = to hold/tenir).  This time there was no OE equivalent for entertain, so a Romance loan word was imported.
So to summarise, PDE undertake does not come from Anglo-Norman entreprendre (mostly "to attack") or even Late Latin imprehendere/interprehendere ("to assault by surprise", "undertake").  However they do have common traceable PIE roots and there is, as you suggest, a correspondence between entre and under in a number of cases.

(1) Ernout Meillet have an interesting conjecture why Latin uses both in and inter (see in)

