Reasons for the construction of overhanging upper storeys are sometimes presented as universal but this type of building is far from prevalent and varies quite distinctly. In most cases, especially in Germany, France and the overhanging upper floors protrude by several inches, usually just enough to accommodate the sill plate that level or a bit more. Remaining examples of the high-medieval buildings show that significantly protruding storeys, like some of the most distinct buildings in Small Shambles in York are an exception rather than a rule.
Did the house owner need to extend their houses? Not really. Average plot in high medieval cities that started to get denser infrastructure with houses often being erected next to one another, leaving a very narrow passage or no passage at all between adjacent building was usually 5-6 meters wide and 18-20 meters deep, meaning that the building could have been always build inwards if necessary. Although the significant part of the plot was used for storage sheds, outhouse and similar utilities, there was always a space to build the house two-three feet longer without the necessity to storeys hanging over the street. Conversely, a lot of buildings erected with a lot of space around, usually located outside the city limits, such as toll houses or rural residences also show characteristic overhanging storeys that cannot be justified with the wish to creatively increase the surface of the building. So it is unlikely that this particular method was invented to increase the living area.
It is very possible, however that the already present cantilevered construction methods were used to increase living space. London Assize of Nuances from 13th and 14 centuries mention the residential houses overhanging the adjacent plots or streets, although the size of the protruding elements were still small, usually not exceeding 2 feet. This seems not a lot, but then again, two feet on all sizes, what was possible to achieve with relatively large lots in 12th or early 13th century century, with the length of the average 13th-14th century house equal to 35 feet and width of 15 feet on a ground floor as per aforementioned source, gives us the area (total, not the living space) of 525 on the ground floor, 741 on the second and 989 on the third (or 10 square meters per storey). Even in cases of houses adjacent to one another in the late medieval cities what limited the extensions to front and rear only, where the numbers above need to be halved, it was the increase not to be scoffed at.
It should be noted, however, that some high-medieval town buildings, especially in the southern Europe had the upper (second) storeys protruding significantly (by 5-6 feet) but these were not cantilevered but rather supported by pillars extending to the sill plate of the second story, thus forming kind of roof for anyone standing directly in front of the ground floor. In other words, we can say that it was the ground floor that was recessed in comparison to the second storey, forming what essentially was a colonnade. This is generally assumed to be a solution resulting from warm climate and allowing craftsmen to work or sell their wares outside rather than indoors while being shielded from the sun at the same time (please note that in the houses of craftsmen and merchants, the ground floor was generally reserved for work, storage and official business, while the living quarters (great hall and bedrooms) were usually located upstairs. This was, however, not a hard and fast rule, as many larger, double-tract houses had both the 'business part' in the front tract (i.e. this was the part one would enter from the street) and kitchen, hall or living quarters located behind it, towards the back of the building. This solution was quite common in late-medieval houses in the Baltic region, i.e. Germany, Poland or Teutonic State. Such division was also sometimes found also in English houses, as evidenced by e.g. rowhouses of Abbey Row in Tewkesbury.
The taxation argument is often brought up, but it usually is not corroborated by the provision of sources. In reality, there was most likely no tax of this kind and even if it existed, the benefits would have been minimal, as the aforementioned gain of the upper storey size was very small to the overall footprint of the building, and the existing sources of that era usually mention 'assessment of value' i.e. general value of the house determined by the number of storeys, and hearths, materials, condition, plot size, additional structures, decorations, plot size etc. Furthermore, in many regions, including Germany and England, the wealth was usually estimated by the combination of declaration and assessment or even declaration only. For example, in many German states, starting with Stendhal in 1285 and many others following suit in late 13th and 14th centuries, the city councils reserved the right to buy a property at a value assessed by them if they considered that the value declared by the owner is too low. This strongly suggest that the footprint of the house could not have been a basis or a substantial part of a tax assessment as it is very easy to measure thus leaving no space for finagling.
From the builder's perspective, limited cantilevered structure can have some structural benefits. With the upper walls being supported by the sill beams that are supported behind the wall, not directly underneath it may reduce the sagging of the longitudinal beams to some extent. This is, however, only feasible with the frame or half-timber construction with the wicker and clay used as a filler due to relatively low mass. This is why jetties and corbelling are most commonly found in the areas favouring wooden construction, while the Italian or southwestern Baltic architecture often made of stone and brick usually lack jettying of any kind save for decorative false corbelling and oriel windows. Jettying makes it also easier to build an overlaying timber-framed storey on the stone or brick walls during the expansion of the existing structure or renovation after the fire or a similar event. This also applies to the possibility that some of the cantilevered structures were the result of building the houses one storey at a time, as in this form of timber frame construction, every level is separate and self-contained with a floor and a roof (what has an additional benefit of providing additional insulation above and below) and it is easy to build the floor on the sill plates slightly protruding beyond the outline of the lower floor instead of painstakingly aligning all elements with the existing outline.
So, to sum it up, the most probable cause of forming an overhanging upper storeys was the ability to extension of the living space facilitated by the popularity of the wooden and half-timber construction that made such a solution possible and beneficial.
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