Basically, measurement level describes certain assumption about the variables at hand. These assumptions imply additional information or structure of the data.
For example, take the interval level. Just as at the ordinal level, values are ordered in a certain progression. In programming terms one could say that the interval class inherits the properties of the ordinal class (not to worry if you are unfamiliar with the concept). In other words, the interval class has the same properties as the ordinal, but an additional property is added. Namely, the distance between the "ticks" on the scale is now equal.
So we know that just as on ordinal level 4 is above 2, but we also know that the distance between 0 and 4 is twice as large as the distance between 2 and 4.
Ordinal methods use the information about the order of values. The same information is contained on the interval level. However, by applying ordinal methods to the interval level you waste the information about the homogeneity of distance between values. Still, it is possible.
On the other hand, interval methods rely on the assumption of distance homogeneity. This is why they cannot be (correctly) applied to data which violate that assumption. Still, this is often done e.g. with Likert scales, but this is a whole other discussion.
Hope this helps.