What did smiths/metallurgists know about why some steel was stronger than other steel in the early 15th century in Western Europe? I want to know so I can determine whether my mages could come up with a spell to increase the strength of steel by removing impurities via magic or not. If they could come up with wording for the spell in the "magic language" they could cast it, but do they know enough to do that?
For a very long time (say, before the 16th century) they did not even know that steel and iron were different materials, and they had only very imperfect hit-and-miss processes to harden iron (by building a thin layer of low-carbon steel on the surface, but they did not know that). (Note: That's in Europe. The Chinese did have cast iron, which is very obviously a different material, and the Indians had a practical technology to make small but consistent amounts of low-carbon steel. Look up Wootz steel; as user Mormacil mentions, Indian steel was an expensive and very desirable commodity, exported to the west to Persia, the Near East, and Europe, and to the east to China.)
Before the development of modern chemistry and metallurgy, nobody had any good idea of knowing why a material had different properties from another material. At best they knew that if they followed a certain process they would get a material with certain properties; around this knowledge, there were weird and wonderful philosophical and even mystical constructions with no practical application whatsoever. (Phlogiston or alchemical transmutation, for example.)
Further musings: steel is not made by "removing impurities". Steel is an alloy of iron and carbon with a well-controlled amount of carbon. To make steel one can start with pure iron and add some carbon (crucible steel), or one can start with cast iron and remove some carbon (converter steel). Before the development of chemistry nobody even knew that iron and carbon were pure elements -- the prevalent theory in the late Middle Ages was that on the contrary the oxides (which they called "earths", hence our term "rare earths" for the metals in the lanthanide series) were the pure elements, and the metals were combinations of an "earth" and phlogiston, the pure element of combustion, a sort of negative oxygen.
I beg to differ slightly from AlexP's answer. Before a modern understanding of "steel" vs. "iron" was fully understood, a fair number of sword makers figured out that, not only were proper amounts of carbon required, but that much more hammering and tempering were also required for the best results (also, sometimes meteoric iron was found and used, which was far superior as a base material than scrape-mined terrestrial iron.) This understanding was first garnered in the Middle East (hence the fame of "Damascene Steel from Damascus.) Furthermore, early accidentally made excellent swords in early medieval Western Europe led to the myths of the great swords (Excalibur etc.) as inferior swords tended to be strong but brittle - meaning they just might crack and literally fall apart in battle.
Alternative solution: It's not usually possible for a human being to do anything well without coordinating their efforts with one or more of their senses. Imagine operating a construction crane with a blindfold on, doesn't work. This means they must receive some sort of feedback from the thing they are manipulating magically. Every action has an opposite reaction after all.
Your Wizards may not know exactly what they need to do from a scientific perspective but they can use their senses to feel out an object and compare it against their experience with "good" or "pure" versions of the same object. They might describe this the same way cooks smell food and can tell if something has been seasoned correctly.
Have them first "feel out" well made swords and then ask them to make poorly made swords more like them. Have them experiment with making swords more this or more that to discover how to make swords surpassing that which can be made by even the greatest blacksmiths in the world.
However, just like cooking, ingredients matter. You can't make a masterpiece with rotten goods. If this magic is based on science/reality, your wizards are going to have to carry around a good deal of material to work with. It may be common to see them traveling around with a cart of vials and powders that they reach into and rub over objects before working their magic on them.