(Not a teacher)
Your brain is processing language at a far greater speed than you are speaking the words. This phenomenon can be seen in things like 'spoonerisms' where phonemes are switched around from preceeding and following words, often to the listener's and speaker's amusement - eg. "Three cheers for our queer old dean!" (rather than 'dear old queen').
Often, the result of our brains rushing ahead of our speech is that we miss sounds out - technical terms like elision, assimilation, coarticulation, lenition etc all occur as a result of this. This kind of speech is gerenally referred to as 'connected speech'.
Tongue-twisters force our articulators to be used in ways that the brain finds hard to cope with in the same way it would with normal speech. The result is that it gets a lot of it wrong.
I must point out that this is all presumption, I haven't actually ever came across a psycholinguistic explanation of tongue-twisters. Seems plausible though, eh?
Basically what I'm saying is that normally your brain controls your articulators - tongue-twisters make your brain overload and lose control.
Student or Learner