Okay. First I'll mention that Buddhism does not use the word "reincarnation." Buddhism uses the word "Rebirth." There's something which gets a Rebirth, reborn. As I understand, reincarnation implies that there is an entity that continually travels, the same entity all the time, reincarnated. That's different to Rebirth. But as to why it is important at the moment of death to have a peaceful mind, I may mention this as a bit of an insurance policy. If you are peaceful you are happy, you are happy here and now. If you are peaceful tomorrow, you have got to be peaceful here and now. If you are peaceful, loving and you develop all these good qualities, then you are happy, each moment and each moment after that. If Rebirth is true, then we would like to be peaceful at the moment of death so that our next moment, our Rebirth, will be peaceful also and based on that - that's our kammic effect.
They also say that your thoughts at the moment of death will push you towards your next Rebirth. So even if you have done a lot of goodness in your life, but when you die you are full of regret, you are thinking of something you did which was not skillful, they say that is going to prepare you for your next Rebirth. Which is not going to be pleasant if you are filled with regret, aversion and all this, you're actually going to get an unbeneficial Rebirth. However, if you only did a little bit of unskillful actions in your life and lot of goodness, then that effect on the moment of Death will only stay in effect for a short period before the good Kamma then fruits. This is all theory here, so take it for what you want, but we teach that it is important to keep a peaceful mind each moment as well as the moment of death. In case Rebirth is true, we, personally, don't want to lose out, so it's up to you.