For the most part assessment was based on photographs. Bombers were equipped with cameras, primarily to assess the accuracy of their bombing, and specialist photo reconnaissance (PR) aircraft followed up bombing raids as soon as conditions permitted to record the results. British efforts were rather haphazard for the first years of the war, but the creation of the Central Interpretation Unit (CIU) in 1941 for photographic interpretation and steady improvements in both numbers and quality of PR aircraft meant that by the time the combined bomber offensive began in earnest there was a mature system in place. K Section of the CIU analysed photographs to assess damage to buildings and calculate total areas destroyed or burned, rapidly producing a First Phase summary within hours followed by a more detailed Second Phase assessment. The results were shared with other departments including the Ministry of Economic Warfare (MEW) and Research and Experiment Department 8 (RE8) of the Ministry of Home Security. The former combined photographic assessments with intelligence gleaned from other sources (refugees, agents, intercepted communications etc.), the latter were responsible for analysing damage caused to British targets by German bombers and resulting loss of production and could thus extrapolate to calculate the impact of British raids. The resulting Third Phase reports were the ultimate assessment of the overall effectiveness of bombing.
Post-war both Britain and the United States also carried out bombing surveys (though the US effort was substantially better resourced) in which teams visited Germany and Japan to conduct more detailed research, resulting in the United States Strategic Bombing Survey (USSBS) and Strategic Air War Against Germany, 1939-1945 - The Report of the British Bombing Survey Unit.
For more detail see the thesis of Robert S. Ehlers Jr., BDA: ANGLO-AMERICAN AIR INTELLIGENCE, BOMB DAMAGE ASSESSMENT, AND THE BOMBING CAMPAIGNS AGAINST GERMANY, 1914-1945.