Why does this DNA extraction work?
Through this procedure you were able to extract the separate strands of DNA out of nuclei of the banana cells. The detergent broke apart the cellular membranes and the salt binds to the DNA to make it less able to remain soluable in water (it neutralizes the charge of the DNA and it is less able to bind to water). Once the DNA was in the banana solution, you were able to use methylated spirits to precipitate DNA out of the solution as DNA is insoluable in alcohol based solutions.
Basics background information on DNA
Deoxyribonucleic Acid (DNA) was first identified and isolated by Friedrich Miescher in 1871, and the double helix structure of DNA was first discovered by James Watson and Francis Crick, using experimental data collected by Rosalind Franklin and Maurice Wilkins. DNA is found in the nucleus of living cells and it is this complex chemical that codes for how living things grow and function, with this code being transferred to subsequent generations.
DNA has a basic structure of a sugar phosphate backbone with 4 base pairs that alternate through the middle of the helix. The standard picture you see is a helical structure as below, however there are several ways the DNA helix is known to spiral, including left and right-handed versions.

DNA model:
Red = Ribose sugar,
White = Phospate
Pink, Blue, Yellow and Green = the Base pairs (Guanine/Cytosine & Adenine/Thymine)
There are 4 base pairs within DNA which always pair the same way, Guanine binds with Cytosine and Adenine binds with Thymine. The amazing thing about DNA is that it's the alternating base pairs that creates the genetic code of every known living thing, from a butterfly to an elephant. Segments, or pieces, of DNA are called "genes". Each gene determines something about our bodies—a trait. In our DNA there are genes that are responsible for hair color, eye color, earlobe shape and so on. Some characteristics, like eye color, are more or less determined directly by DNA whereas other traits are determined both by DNA and by your environment as you grow up such as how tall or short you become as an adult (due to potential differences in diet). DNA is constantly maintained in cells by a number of enzymes, allowing for insertion errors and random mutations to be removed however not all of these are caught and as such the DNA slowly changes over time (helping to cause aging in the process).
Every human shares 99 percent of his or her DNA with every other person, most likely due to a major event that nearly brought the human species of the brink of extinction thousands of years ago. We share most of our genes with fellow primates such as chimpanzees and with other mammals such as mice. We even have genes in common with the banana plant!
Further reading
Genetics is a very complex subject and a simple DNA extraction experiment is only just starting the journey! We recommend you spend some time reading up on some extra sites to further your knowledge:
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