"Whoever tries to keep their life will lose it ..." (Luke 17:33-37)

"Whoever tries to keep their life will lose it, and whoever loses their life will preserve it. I tell you, on that night two people will be in one bed; one will be taken and the other left. Two women will be grinding grain together; one will be taken and the other left. "Where, Lord?" they asked. He replied, "Where there is a dead body, there the vultures will gather." (Luke 17:33-37)

Is this about the time of death?

Jesus continues to discuss the time of death, at which time those who have dedicated their lives to following the teachings of Jesus will be taken back to the spiritual realm. Those who have focused their lives on the temporary things of this material world will, according to Jesus' teachings, be left behind. In other words, they will remain in this hellish material world.

This point is confirmed when Jesus' disciples asked Jesus:
"Where, Lord?" (Luke 17:37)
Whereupon Jesus replied:
"Where there is a dead body, there the vultures will gather." (Luke 17:37)
Remember that Jesus had just said, "one will be taken and the other left." So the question is "where" are the others left?

(Some early manuscripts add a verse, 17:36, which is not included in the New International Version because it is not in the earliest Greek texts. Here is the verse, from the New King James Version:
“Two men will be in the field: the one will be taken and the other left.” (Luke 17:36 NKJV))
The point remains that Jesus is speaking of some being taken back to the spiritual realm while the others will be left behind - here in the material world - "Where there is a dead body, there the vultures will gather."

Why do vultures gather?

The reason vultures gather around a dead body is in order to eat the carcass. And the reason a person remains in the material world and isn't taken back to the spiritual realm after death is due to a combination of desire and consciousness. A person who is attached to the world of matter and desires to enjoy the world of matter will remain within the material dimension.
So how can a person "remain" in this material world after the physical body dies?

We are forced to take on another physical body.

While many might call this reincarnation, the word "reincarnation" is often mistaken to mean that I - this physical body - will take on another physical body.

But this makes no sense. How can a physical body take on another physical body?

The reality is that the "I" here - each of us - is spiritual in constitution. None of us are these physical bodies. We are each driving this physical body much as a driver drives a car.

A car doesn't move until the driver sits in it and drives it away. In the same way, our physical body is lifeless unless it is being driven by the self within. And once that self rises out of the body at the time of death, the body becomes lifeless. This fact is confirmed by the description of Jesus' healing of the Jairus' daughter:
They laughed at him, knowing that she was dead. But he took her by the hand and said, "My child, get up!" Her spirit returned, and at once she stood up. Then Jesus told them to give her something to eat. (Luke 8:53-55)
"Her spirit returned" clarifies that the spirit gives the body life and when it leaves, the body is dead.

Where does that spirit-person go after leaving the body at death?

This is what Jesus is discussing. Those who follow his teachings and reawaken their innate love for the Supreme Being will return to God in the spiritual realm.

But those who are attached to the material world and the things of this world - will be left behind to take on another physical body in this hellish world - where "the vultures will gather."

Yes, this is indeed hell. But there are higher and lower planes of hell. A person living in a wealthy country in a wealthy family might be living in less of a hell than a person living in a war zone in the Middle East for example. Or a person starving in a poor country in Africa.

Did Jesus and Jewish society accept transmigration?

Jesus and much of Jewish society at that time accepted the transmigration of the spirit. This was communicated multiple times in the New Testament, including the fact that many believed Jesus himself had transmigrated:
Once when Jesus was praying in private and his disciples were with him, he asked them, "Who do the crowds say I am?" They replied, "Some say John the Baptist; others say Elijah; and still others, that one of the prophets of long ago has come back to life." (Luke 9:18-19)
The only way Jesus could have been these people "come back to life" would be if the spirit of one of these prophets had transmigrated into the body of Jesus. Since the prophet's body had died - some centuries earlier - and Jesus' body was not living at that time - they were accepting that the spirit could transmigrate from one body to another. This was also communicated by Jesus' disciples asking Jesus this question about a man who was born blind:
His disciples asked him, “Rabbi, who sinned, this man or his parents, that he was born blind?” (John 9:2)
Jesus' disciples accepted that the spirit of the man lived in another body prior to being born. How else could his sins have caused his blindness?

Essentially, to transmigrate into another body - of a human or animal - is what Jesus is referring to being "left" in Luke 17:35. Being left in this context means to continue the suffering and pain inherent within the hell of the physical world. This is accomplished by taking on another gross physical body at some point after the death of the current body. This provides the substance of Jesus' statement in Luke 17 above.

Jesus doesn't want us to remain in hell - where we constantly struggle for survival and suffer diseases, pain, aging, and ultimately the death of this body. Jesus wants us to renew our loving relationship with the Supreme Being and return to our eternal nature in the spiritual realm. This is why Jesus' most important instruction was:
"'Love the Lord your God with all your heart and with all your soul and with all your mind.' This is the first and greatest commandment." (Matt. 22:37-38)

How can a person keep their life by losing it?

The reason Jesus uses the juxtaposition of preserving one's life by losing it ("Whoever tries to keep their life will lose it, and whoever loses their life will preserve it.") relates to giving up the attractions and attachments of the physical world.

By putting our love upon the Supreme Being we will be able to gradually give up or lose our attachments and attractions to the physical world and thus lose our notion of life in this physical world. But in return we will gain or preserve real life - the life that sustains us spiritually: Love and loving service to our Best Friend the Supreme Being.