Answer: The double E and double U indicate that the letter represents a plural. Some other common Spanish abbreviations, among them FF. AA. for Fuerzas Armadas (Armed Forces) and AA. EE. for Asuntos Exteriores (Foreign Affairs), do the same thing. (Also in very common use are the abbreviations without the spaces and/or periods, such as EEUU, FFAA and AAEE.) Such a doubling of letters isn't done for all plurals; ONU is the abbreviation for la Organización de las Naciones Unidas, the United Nations.
We do the same doubling of letters in English in a few cases for words of Latin origin. For example, the abbreviation for "page" is "p.," while for "pages" it's "pp." (The same abbreviations are used in Spanish for página and páginas.) And the abbreviation for "manuscript" is "MS" or "ms," while in the plural it's "MSS" or "mss." (Again, the same abbreviations are used in Spanish.)
You will in Spanish occasionally see the abbreviations EUA (for Estados Unidos de América) and even USA for Estados Unidos, but much less commonly than EE. UU. and its variations.