Methanesulfonate is a very happy anion, which is one of the reasons why methanesulfonic acid is used to make salts of basic drugs. The ions of these formulations tend to not be too tightly bound or paired, making the salt forms generally easier to dissolve. [The stronger the interactions between the ions, the harder it is for water to break up the party and dissolve them].
However, the contentedness of the sulfonate anion makes it a good leaving group when it is part of a covalent molecule. It would rather be off floating around with a negative charge on it again than be tied up in a regular bond, so sulfonate esters and the like tend to be pretty reactive.
Alkyl mesylate esters, then, are also pretty toxic. Just how toxic is the question, though. The stuff will react with nucleophiles wherever it finds them, and if that is some protein on the surface of a soon-to-be-shed epithelial cell, then no harm done. But there are many other situations that won't work out so well, going all the way up to DNA damage.
So how did this nasty stuff get in there in the first place? When I heard about the contaminant, my first thought was that someone had been washing out the reactors where the salt was formed with ethanol, and that appears to be exactly the case. Alcohol plus free acid under strong acid catalysis will give you ester, in what's literally one of the oldest reactions in the book.
Roche appears to have been able to keep the contaminant down below regulatory levels normally, but someone's mind has been wandering over in Switzerland. In consequence, the fearsome Swiss reputation for purity and consistency takes a torpedo below the water line.