I'll try to answer this by using an analogy:
English and German seem like two very different languages. Yet if you look a little closer you'll find similarities:
- Mother - Mutter
- House - Haus
- Boat - Boot
- Earth - Erde
- Weather - Wetter
- Summer - Sommer
is this just pure coincidence?
The more similarities you find, the more likely it would seem that these two languages have a common ancestry.
"Universality of DNA" means that the blueprints of all life forms on earth is written in the same language.
An example is Synthetic Insulin for diabetics: basically, the specific gene for human insulin is embedded into a bacteria's genome, and because the code is written in the same language, the bacteria 'builds' human insulin, which we can later harvest.
Another simple example is the common cold: the virus infects our cells, injects its DNA (or RNA), and our own cells end up replicating the virus because it's blueprint is written in the same language.
Like with English and German one might wonder "Is it just pure coincidence that humans, animals and plants have developed the same language?"
Or does this hint, that all life on this earth is somehow connected? Maybe through common ancestry?
EDIT: just saw that 'shellholic' posted a similar answer.