Just to complement Qrrbrbirlbel's excellent answer, some comments about why this happens.
" character should never be used in LaTeX to get quotes. The correct way to get (British style) double quotes is
``Opening and closing quotes''
The fact that
" produces seemingly correct quotes (”) should not be relied upon: it can fail depending on the font one is using. Moreover, in good typography, opening and closing quotes should be different from each other.
When different typographic traditions are followed, the quotes change; German uses lowered opening quotes and raised closing ones, but reversed with respect to British usage. The same seems to hold for Danish, and
babel provides shorthands for getting the correct characters without hunting for them on the keyboard, as explained by Qrrbrbirlbel.
In many language modules of
babel the “straight double quote” character is used in the same way to introduce shorthands. These work in
babel always in the same way:
Some of these combinations are meaningful (they may mean different things in different languages), others don't. When the combination doesn't have a predefined meaning,
<shorthand_char> fallbacks to something that, in the case of
" is “print the character in the current font at position 34”. However the procedure for recognizing the following character uses
\@ifnextchar which, by design, eats all the spaces that it finds in the process.
So in the example
Hello "I AM A" DUMMY SENTENCE
LaTeX looks whether the combination
"I is meaningful for Danish, which isn't, so it prints
Then it finds the second
" and starts looking for the following character, swallowing spaces. Eventually
A is found and since
"A is not a predefined combination,