You surmised right about the 15 gianfri, which is the rim dia the tire is made to fit on expressed in inches (yeah, I know the units are a mix of imperial & metric, but that's just the way it is!?) The 225 bit of 225/50R15 is the width of the tire body in millimetres, not necessarily the tread width altho it's going to be fairly close; while the 50 is the sidewall height expressed as a percentage of the 225mm. Similarly, the 165 of 165/55 is width in mm, with 55 being the sidewall height as a percentage of the 165mm.
Rear tires can't go much wider, due to the belt & sprocket, but they can go a little taller, ie, 225/60R15's will generally fit on an RT - the only increase is sidewall height, but that does mean the rolling diameter of the tire will generally be a little larger. For that reason many drop the width, ie, go for a 215 or even a 205, as well as a taller sidewall height, ie 60 or possibly even 65, in an attempt to keep the rolling diameter close to the same as the OE tire. Usually, you'd select a 215/60 or a 205/65, but just to make things a little more difficult, the tire sizes shown on the sidewalls are only NOMINAL, and very few tires with the same sidewall nominal size are actually physically the same size.... they are just somewhere about that nominal size, altho the differences can be as much as 20mm or more difference in rolling diameter!!
Front tires can go a bit wider but not much taller, the max for the 15" rims generally being about a 175/60, altho there are a few who've managed to find & fit 185/50's or even a (small) 185/65!! Still, I'd work to 175/60 being about as big as you'd want to try without a test fit, and even so, you might need to ease the lip of the fender over the tire during fitting, just to get it on - once on, they usually run OK if the fender supports aren't bent, altho some have swapped all the hex head screws under the fender for dome type phillips head screws to minimise the chance of the screw heads rubbing on the tires as they rotate.
Why go bigger, apart from finding a tire to fit that isn't the OE stuff? A slightly larger tire size will generally have a slightly larger footprint on the road, and it's that footprint that provides traction. A larger area can mean better grip, especially if the footprint has increased proportionally more in length (along the line of travel) than it has increased in width, but that traction increase & to a certain extent, the longer more than wider footprint generally only occurs if the tire's pressure is a little lower, and that's a whole 'nuther discussion. Still, if the footprint is a longer than it is wide footprint, you'll generally get better traction/grip & directional stability and depending upon the wheel alignment being correct, you can also get a longer tread life to go with that better grip. The risks are that a less efficient tread pattern on a larger tire (especially wider) can mean there's slightly more propensity for hydro-planing, but once again tire pressure plays a big part in avoiding that; and a longer &/or wider footprint will make for a little more steering effort, but with the increases in length & width we are talking about here, you'd need some fairly sensitive equipment to measure it, unless there were other issues at play (like poor wheel alignment for one...)
The slightly narrower but taller profile rear tire can mean marginally less outright traction on dry roads, but a good wet weather tread pattern can counter that minor issue by providing noticeably better traction, especially in wet weather & less propensity for hydro-planing on wet roads; and the slightly larger rolling diameter will not only help make the (by manufacturers design) up to 10% optomistic speedo/odo readings more accurate, bringing them to within 1 or 2 mph/miles of actual instead of reading over as they do, but it will also help the tire achieve a longer tread life & have less of a tendency to 'throw the center of the tread out whole rotating' so the tire shouldn't wear the center of the tread out anywhere near as much as the OE Spec tires do!
Help any gianfri?