Excerpt from an online etymology dictionary (etymonline.com)
The use of the word in radio communication to mean "yes, I understand" is attested from 1941, from the U.S. military phonetic alphabet word for the letter -R-, in this case an abbreviation for "received." Said to have been used by the R.A.F. since 1938.
The military no longer uses Roger to mean 'R', it now uses Romeo
Early wireless transmit and receive communications were unclear and subject to atmospheric distortions, so clear diction was imperative. Words were pronounced to end in -er for clarity, as in oner, twoer, three-er etc. Roger was a common name then which ended in -er and was adopted to mean "Received and understood".