MRSA is spread by contact. So you could get MRSA by touching another person who has it on the skin. Or you could get it by touching objects that have the bacteria on them. MRSA is carried, or "colonized," by about 1% of the population, although most of them aren't infected. That quote is from the WebMD link.
I don't think disinfecting the schools is doing any good at all. Perhaps, they are thinking they have to be seen doing something.