How do you explain "back in the USSR"?

You don't know how lucky you are, boy...

For the same reason as "The UK" or "The USA" - all of them are acronyms starting "The Union of ..." or "The United ...".

Similarly you say "The commonwealth of Massachusetts" but not "The Massachusetts".

This does not apply to Ukraine, unless you want to say "The Republic of Ukraine".

... or "Born in the USA"