Luke 17:34-37
34 I tell you, in that night there shall be two men in one bed; the one shall be taken, and the other shall be left.
35 Two women shall be grinding together; the one shall be taken, and the other left.
36 Two men shall be in the field; the one shall be taken, and the other left.
37 And they answered and said unto Him, "Where, Lord?" And He said unto them, "Wheresoever the body is, thither will the eagles be gathered together."
(KJV)
The answer to what the one taken and the other left is about is given in that very last verse. Christ gave a parallel verse to that in Matt.24:28 where the word "body" is "carcase" instead (literally per the Greek also). The first one taken is taken in deception. Eagles do not... go after what's already 'dead' (carcase). So those eagles are really FAKES, trying to pass themselves off as authentic. In the going after 'carcase' sense, those are actually vultures.
The Luke 17 event of two men in one bed, two women grinding at the mill, two men in the field, one taken and the other left, is about the deceived vs. those who remain waiting on Christ.
Per Christ's parable in Matt.13, the 'field' represents the world. So with the two men in the field, one seeks to be taken out of the world, and the other does not.
With the two women grinding, that's about working. To stay at the grinding wheel is symbolic of working for Christ, remaining faithful all the way to His coming, not being deceived. It's about remaining a good steward, giving the meat in due season. The one who seeks to leave goes to another, not waiting on Christ.
This 'taken' idea was actually first covered back in Isaiah...
Isa 8:13-17
13 Sanctify the LORD of hosts Himself; and let Him be your fear, and let Him be your dread.
14 And He shall be for a sanctuary; but for a stone of stumbling and for a rock of offence to both the houses of Israel, for a gin and for a snare to the inhabitants of Jerusalem.
15 And many among them shall stumble, and fall, and be broken, and be snared, and be taken.
16 Bind up the testimony, seal the law among My disciples.
17 And I will wait upon the LORD, That hideth His face from the house of Jacob, and I will look for Him.
(KJV)
Notice how those 'taken and snared' is given in contrast to those who 'wait upon the LORD' and look for Him.
Isa 28:13-16
13 But the word of the LORD was unto them precept upon precept, precept upon precept; line upon line, line upon line; here a little, and there a little; that they might go, and fall backward, and be broken, and snared, and taken.
14 Wherefore hear the word of the LORD, ye scornful men, that rule this people which is in Jerusalem.
15 Because ye have said, "We have made a covenant with death, and with hell are we at agreement; when the overflowing scourge shall pass through, it shall not come unto us: for we have made lies our refuge, and under falsehood have we hid ourselves":
16 Therefore thus saith the Lord GOD, Behold, I lay in Zion for a foundation a stone, a tried stone, a precious corner stone, a sure foundation: he that believeth shall not make haste.
(KJV)
Because God's Word was to them "precept upon precept, precept upon precept, line upon line...," His Word became a stone of stumbling and rock of offense to them, so they might go and fall backward, be broken, snared and taken (i.e., into deception). Notice how God contrasts those with the believer that stays on the true foundation through Christ, not making "haste". The idea of haste means to be in a hurry. We can't be that way if we want to understand God's Word, nor can we be that way about Christ's second coming in wanting to escape and leave.
Per Matt.24:23-26, 2 Thess.2, and Rev.13:11 forward, Christ revealed that a specific false one is to appear in the last generation that will see Christ's second coming. That false one is to work great signs and wonders on the earth to deceive with, exalting himself over all that is worshiped or that is called God. We're told that false one will setup an "image of the beast" for all to bow in false worship, with those who refuse are to be killed. That's who those fake "eagles" which the first ones taken will flee to. Those fake eagles represent Satan's host on earth with their god, the coming false messiah. The deceived first ones taken will think that false one is Christ having returned.