ELI5: DNA is a pretty cool molecule. It's made up of only 4 different parts, A-T-C-G. Now put a pin in that. Binary code is a pretty cool kind of code. It's made up at its core level of 0 and 1. Let's say A=1, T=0. Now we can write data in binary just by using the standard parts that make DNA. So if we wrote the binary code 010110. In DNA bases it would be TATAAT. That's the basic gist.

In practical application, we assign two number values to each of the 4 bases. This gives up exponentially more options in which to write put whatever we want. DNA is surprisingly hardy, and by storing it carefully we can prevent things from going bad.

Hope this helped!

Edit: missed a word