He's not entirely correct, in the first place. The labeling on multi-grade oils shows two viscosity numbers. The first one, the one with the 'W', is the "winter" viscosity, which is not exactly what that is, either. What it means is that when the oil is at 70 degrees, it has the same viscosity as an SAE 5 weight engine oil. The second, higher number is the comparative viscosity at 200 degrees, so at that temp, a 5W-40 flows and pours the same as a single grade 40 weight would. So does a 10w-40 and so does a 20w-40. At operating temperature, there is no difference. So he's mostly wrong.
The advantage of multi-grade oils is that they flow much more readily when cold, which can be very important on cold starts, particularly in really cold weather, although a 10w-40 will work perfectly well in temps down, to around 20-25 degrees F.
But what is a factor has to do with how multi-grade oils are made. The oil actually starts at the lightest grade indicated, so when making a 5w-40, the blender starts with a base oil stock that is SAE 5wt at 70 degrees. A straight parafin base petroleum oil at this weight would actaully be a 5w-5 in practice, much too thin to use, so additives known as Viscosity Index Improvers (VII's) are used. These are long chain polymers that become bulkier when heated, thickening the oil so that the result is that heat does not thin out the lubricant as much as it ordinarily would. Enough VII's are added to get the desired result, in this case, a 5wt oil at 70 that is as thick as a normal 40 wt when heated, labeled as 5w-40.
The problem comes when the oil is used in a transmission. The gearbox imparts a very heavy shearing force to the oil, and works to physically shred the large molecules of the polymer additives so they no longer prevent the thinning of the oil with heat, and you can end up running a oil that is really a 5w-20 in a surprisingly short time (as little as 2 hours run time) if the oil was an inexpensively blended lube intended only for use in engines. The better oils use more expensive, but much tougher, polymers that were created for use in multi-grade gear oils, and the really good oils made today will last 4 to 5 times longer without falling below their rated weights.
Most true synthetic base stocks (primarily poly-alpha olefins and esters) have a naturally higher viscosity index than natural pertoleum oils do, so a good PAO oil that is 5 weight cold may act like a 10 or 20 weight at 200 even without modification, but it is nonetheless true that the lower the "winter" oil weight, the more the oil will depend on the addition of VII's to make it a 40 wt at 200, and that means that it will be more subject to viscosity loss due to additive degradation from shear. A 5w-X, then, is more likely to fall out of grade early than the same type and quality of a 10w-X. So he's partly right, in some cases.