Humming is all about vibration, the same as when you hum a song. If there is any room for the plates (especially, but also coils or anything else) to vibrate, they will make noise, as the electromagnetic forces in the transformer push and pull on them. Reduce the scope for vibration, and you reduce the hum.
1. Sometimes all that is needed is to tighten up the bolts that hold the plates together. They may have loosened over time. Torque them down and the hum will be reduced and it may disappear.
2. Tightening down the bolts that hold the transformer to the chassis may help, too. This is analogous to tightening down a driver in a speaker case, or a tonearm on a plinth; the chassis may act as a vibration "sink".
3. Alternatively, depending on the forces involved, you might try isolating the transformer from the chassis. It's possible that the chassis is acting as a resonator for the vibrating transformer, much as a guitar or violin body acts as a resonator for a vibrating string. In this case, isolating the two using something like the wooden board Tubed mentioned above, might be the answer.
4. Re-impregnating with new varnish, as Hyperion suggested, can work for exactly the reason Ohighway suggested: it fills up the gaps and seals the plates together, so they can't vibrate (move) against each other.
5. Potting the transformer --embedding it in a case filled with tar (or similar deadening material, such as a wax formula or even heavy oil)-- can also dampen vibrations, and thus silence hum.
Hum arises as the old varnish dries, shrinks and crumbles, and as the bolts holding things together loosen over time. Even repeated expansions and contractions from large temperature changes can loosen things up. Any of the above steps, that serve to remove that looseness, will help eliminate hum.
NOW, THE REAL EXPLANATION:
The REAL explanation for hum is that the electrical gremlins begin to giggle, when they are tickled. Actually, these particular gremlins are of a special type, correctly called gremlams, or just "lams" for short. That is where the term laminations comes from: the laminations are where the gremlams congregate. :yes: If you are familiar with the Lamm brand of audio amplifiers, their name is a subtle reference to the larger-than-usual number of gremlams found inside their transformers, the extra "m" indicating up to x1000 more lams inside.
Like most types of gremlins, lams are very sensitive to substances; they like to be drugged, and behave best when they are! Flowing electrons act as a drug on them. Normally the flow of electrons in the trafo keeps them dulled and idle. It is like Ritalin or Morphine times 100 for these particular gremlams. Usually, when the current is flowing, they are happily dreaming bizarre dreams (I won't get into the contents of those dreams; this is a family forum and it isn't good for people to know too much, but take it as a hint that their excitement while dreaming is part of the reason a transformer can put out more voltage than you put into it!).
The reason plates are kept tight in transformers is that it keeps the gremlams happily squished up in nice rows, like sardines in a can, but when plates get loose, there is room for one or another of them to wriggle a bit and even to roll over in his sleep, in response to the dreaming. When one rolls over, he inevitably rubs the one next to him a little bit, which tickles that one, causing him to giggle... the shaking motion of the giggling tickles the one next to him, causing THAT lam to giggle, too, and so on. Soon, enough gremlams are giggling in unison, which we hear as hum.
The looser the plates, the more violently the lams can wiggle and giggle, and thus the louder the hum. Tighten the plates back up, and they don't have room to roll over, wiggle or giggle, so no more tickling and no more choral giggling (= humming).
Likewise, feed them varnish and they get so high they just lie there, basically comatose from intoxication, too stoned to giggle or hum. 
Cover them with tar and they lapse into a state not unlike a bear's estivation. Different routes to the same effect, best known as...
The Silence of the Lams.
