Short answer? They didn't, at least not deliberately targeting culturally rich areas. They did attack civilian areas and population centres, but given the structure of industry in the era, the limitations of strategic bombing accuracy and the theory behind the campaign these were all envisaged as militarily-motivated (or at least politically/economically-motivated attacks contributing to the war effort) attacks. The RAF did seek to destroy housing with the justification that a worker without a home is a displaced person and not useful to the war effort, and that refugees would also cause a drain on the German war economy, diverting resources from the war alongside the theorised morale effect.

As a response to this, German propaganda sought to portray the Allied campaigns as petty terrorism (there is a rather famous poster of a skeleton riding an allied bomber through the night sky, with a title branding Allied bomber crews "terror-flieger"). This extends to trying to alter the narrative around specific notable events - such as the bombing of Dresden. A lot of civilians did die in the bombing - somewhere between 20-30,000 according to records of the Dresden city council - but as part of his work the Minister for Propaganda, Goebbels set about falsifying a much higher death toll to portray the attack as an assault on the German people and their culture. However, Dresden under the laws of war at the time was a valid attack as international conventions only banned the bombardment of undefended cities and there were valid military targets in Dresden, a large railway depot connecting logistical lies for much of Germany's Eastern Front and a large number of factories and workshops producing war materiel. The debate surrounding Dresden won't end any time soon, but the pop-history view of the attack does differ from the (still terrible) truth.

In general, his pattern holds for other cities attacked, facing the inability of weapons of the time to effectively strike the specific industrial targets themselves in a manner that actually prevented their function the RAF resorted to a strategy of damaging the city itself in an effort to deny these industries the support they required to function - workers, clear access, power and the transport to/from them of raw materials and finished goods.

Combined with other factors such as campaigns to reduce German access to resources the overall effort did work, and Nazi Germany's production of war materiel plateaued in 1944 despite efforts to increase it in contrast to 1942-43 where in increased despite US and UK strategic bombing efforts. Whether this was worth the cost in lives depends on your personal view of the culpability of the individual for the acts of the state, the depth to which one considers WWII a 'total war' and whether you accept the premise that the bombing campaign caused a swifter end to the war.

In the book "Churchill, the last lion " Churchill actually began bombing civilian quarters in late 1940 early 1941 in response to the Germans bombing London and other civilian centers. He was quite vitriolic about killing as many Nazi's as possible, and retaliating for the bombing of his people. "Bomber" Harris had reported to him that they could not avoid bombing civilians because their night bombing accuracy was abysmal (his original directive was no civilian deaths). All this tit-for-tat started when german bombs were accidentally released on the east London docks because they couldn't locate their actual target. Churchill took it as intentional and bombed Berlin in retaliation.