This (line) represents man, and this (line) represents the end of his life, and while he is as such, the closest line overtakes him

In both Hadīths, there is an example that illustrates man's hope, his lifespan, all the things to which he is exposed, and his death as a result of one of them. But, if he escapes all of them, then his death will come to him when his lifespan ends. These lines represent the various afflictions that confront him, so if he avoids one, he cannot avoid another, and if he escapes all of them safely, without being afflicted with a disaster such as illness or loss of property or otherwise, then surely he will be overtaken by death. That is, if one does not die due to a cause, then he will surely die as a result of the expiration of his lifespan. So, while a person is in such a state (among these afflictions), the closest line will come to him, which is the end of his life.
The Hadīths imply encouragement of having less hope in the worldly life and preparing oneself for the suddenness of death. The verb 'snap (at)', which in Arabic describes the sting of venomous creatures, is used as a metaphor to emphasize fatal affliction.