Horseshoes are meant to protect the hooves from the kind of damage that would have happened on a long-term basis - in other words, chronic wear, uneven abrasive damage and material fatigue. However, this kind of damage is not something that would immediately become evident at instant after the horseshoe was to be removed. Horse hooves are made of a tough protein called keratin - it's the same protein that hair and nails are made of. It's resistant and reliable enough in, so to say, "native" settings - these that horses have evolved to live in. It's just the horse domestication that brought a different environment, the one that evolution wasn't exactly prepared for.
Wild horses usually tend to travel constantly, covering great distances, but do so at a slow pace - in that case, the hooves are naturally worn in a smooth, gradual and even manner. Domesticated horses, however, often travel at a much faster pace and do so in inconsistent bursts, separated by periods of complete idleness, which makes the wear unnatural and destructive to the whole structure of the hooves. Also, wild horses live naturally in semi-desert climate, whereas the domesticated ones are often subjected to travelling in moist and colder conditions - by which the keratin is significantly softened and rendered prone to damage.
Also, walking without horseshoes wouldn't be painful nor anything like walking on sharp rocks / glass because - like our hair and nails - hooves are composed of inanimate matter. For the same reason, it's possible to painlessly attach the horseshoe to the hoof in the first place. But if it was the case that just one of the four horseshoes got deattached or broken, it would certainly mess with the horse's balance a bit, cause unhealthy muscle strains and uneven load on skeletal structure. Please think about how would it feel to walk or run if you were to wear a thick-sole shoe on one foot while leaving the second one bare.
However, while keratin is relatively hard and tough, steel and aluminium - the most common materials of which horseshoes are made of - are even harder and tougher. And walking without horseshoes in damp and cold environment, especially with a heavy load, would progressively damage and weaken the structure of the hooves and ultimately destroy or split them apart. At that moment, the living tissue underneath would get exposed - and it's when it would start to be really painful.
Regarding whether it is common for the horseshoes to break - no, it's rare but still could happen if the horseshoe is old and its material is severely fatigued due to accumulated damage of its crystal structure at the microscopic level. It is much more common that the horseshoe just slowly becomes loose after a month or so, and needs to be reattached.