Oh, this is easy. This one I was taught many many times. Here is how it goes:
Regular untrained people live their life as if they will live forever. They 1) worry about small unimportant stuff, get offended, or scared, or enraged at things that are not really important. And 2) they waste days, months and years of life waiting for better life.
However, 1) Everyone dies, sooner or later. Even the Buddha had to die. First, our parents will die. Then our friends of same age will start dying. Then the day will come when our breathing will stop and we will die too. Every day we get a day closer to death. Every time we celebrate our birthday, that's one less year we have left to live. The time seems to go slowly, but if you look back you realize it did not take you that long to reach your current age. Similarly, one day you will look back and realize you have spent all of your time.
Also 2) Death comes without a warning. The other day my friend (a doctor) called me and said his patient died in a car accident. He had a big house, 3 children and a housewife. As he was coming home from work, a truck made a mistake shifting lanes, and pushed his car off the road. We have no idea how we'll die. We hope we'll die from old age, in our bed, surrounded by grandchildren - but there is no guarantee it will be that nice. People die from all kinds of causes: from accidents, to disease, to terrorism, to allergic reactions, to poisonous snakes or insects, to plane crashes, to sudden heart attacks, to cancer.
So 3) We should stop expecting we'll live much longer. There is absolutely no guarantee that we will live to see the next summer, or next Christmas, or our next birthday. In fact, we should get into habit thinking that any day could be our last day on Earth. When we go to work, we should expect that we may not come back. When we go to bed, we should expect that we may not wake up.
Therefore 4) We should not live a lukewarm life. We should live our life as if this was our last day. This pertains both to our Dharma practice, but also to our attitude to life in general. We should perform every act, every conversation as if this was our last battle on earth.
And 5) If we feel like we worry about something too much, we should compare this problem with death. If the problem is worse than death, then we are allowed to worry. But if the problem is not as bad as death, then we should sober up and stop worrying too much.
Mindfulness of death gives us a useful perspective in which truly important things matter and non-important things are seen correctly as unimportant. It also gives us strength to act with courage and not live lukewarm life. So no matter how you look at it, it is a very useful perspective to cultivate. And the way you cultivate it is by thinking about it, meditating on it, until you feel it, until it is with you at all times - judging all of your action and inaction. Then the death becomes your solemn friend: "You don't have that long left. Use it wisely"