Among all of the ancient people of the Bible, the Egyptians seemed to have made the greatest effort in preserving their dead. While the Israelites did very little to prevent nature from taking its course, the Egyptians turned mummification into a science. They actually became very good at it.
I recall seeing years ago a number of Egyptian mummies on display in a museum. There seemed something incredibly bizarre about seeing those mostly still-wrapped (bare withered hands and parts of faces could be seen on some of them) bodies of people who had been dead for thousands of years, laid out in large glass display cases. There are many other examples in museums throughout the world (other example in photograph).
In all likelihood, the Ancient Egyptians discovered mummification by accident due to their hot, very dry climate which would have very well preserved some bodies naturally, without any human involvement. From there, they would have experimented with various methods and substances until they reached a very high level of expertise.
The apparent primary reason that they had for preserving their dead in such a way was religious. They believed that the dead would be resurrected one day, and that the old body would be used. They had the resurrection part right, but there was (and is) no need to worry about using the old body again. After all, why be given back the same body that obviously didn't work anymore?
According to the apostle Paul:
"But someone may ask, "How are the dead raised? With what kind of body will they come?" How foolish! What you sow does not come to life unless it dies. When you sow, you do not plant the body that will be, but just a seed, perhaps of wheat or of something else. (1 Corinthians 15:35-37).
And further:
"So it will be with the resurrection of the dead. The body that is sown is perishable, it is raised imperishable; it is sown in dishonor, it is raised in glory; it is sown in weakness, it is raised in power; it is sown a natural body, it is raised a spiritual body (1 Corinthians 15:42-44)
While of course a good Christian funeral service for those who have passed away is always desirable whenever possible, it is not vital to the afterlife. God has better ways of taking care of that.
Fact Finder: Jacob/Israel was embalmed (mummified) by the Egyptians. How many days did the process take? (a) 3 (b) 7 (c) 25 (d) 40 Genesis 50:2-3