How can Muffliato be used in a crowd?

You are absolutely right: These two things do not fit together. Here is the definition of Muffliato (one of the spells the Half Blood Prince invented) when it is first used:

Harry had already attempted a few of the Prince’s self-invented spells
...
and, perhaps most useful of all, Muffliato, a spell that filled the ears of anyone nearby with an unidentifiable buzzing, so that lengthy conversations could be held in class without being overheard.

^{*Harry Potter and the Half Blood Prince, Chapter 12: Silver and Opals}

When I read your question, I would have thought, the spell just "buzzes" the words that the caster speaks and so it would be just a noise and people would still be able to hear anything else (e.g. the teacher in the classroom). But the definition definitvely contradicts this.

But then again there is this occurrence:

Harry pointed his wand hastily at the door of Madam Pomfrey’s office and muttered, “Muffliato!” so that she would not come running. Then he scrambled to the end of his bed for a better look at what was going on.

^{*Harry Potter and the Half Blood Prince, Chapter 19: Elf tails}

So you can direct the spell to affect specific people. But then again: Wouldn't an experienced witch (teacher) get curious, if she suddenly could not hear anything except a buzzing noise anymore?

I think the definition as given in the book is incomplete and the buzz only buzzes the word of the caster, but there is no evidence in the book, I scanned every occurence in Book 6 and 7 (they use it to protect their tent when on the run) and could not find anything regarding that.

The only explanation why nobody recognizes the buzz could be: It was invented by the Half Blood Prince, so probably nobody knew about it.

In Chapter Thirteen of Deathly Hallows it seems that Muffliato creates silence with no sound:

There was no bustling or talk or swift footsteps here; the purple-carpeted corridors were as hushed as though the Muffliato charm had been cast over the place.

If this is the case then it would be understandable how other students could hear the teachers. There would simply be magical silence emanating from Harry's area, and that would not interfere with the sound of the teacher speaking.

Of course, this does not seem to fit with other descriptions of the Charm which mention a "buzzing" sound, so this is at best a partial answer.