It depends on what the plural of the word ends with. If the plural only adds an "-e", "-s" or the word doesn't change in the plural form, then the definite plural form is "ene" (e.g. Huse to Husene, sko to skoene and Hotdogs to either hotdogsene or hotdoggene)
If the plural adds an "-er", then "-erne" is the definite suffix (e.g. Piger to Pigerne, personer to personerne, etc.)
Also, with things like nationalities and jobs, the pattern tends to be, for example, arbejdere to arbejderne or danskere to danskerne
Simply because the verb "at læse" means also that: http://www.ordbogen.com/opslag.php?word=l%C3%A6se&dict=auto
It's more like studying (also in British English, you can use reading if it's in higher education) for a degree than the act of sitting down and studying. "Læser" is generally used for higher education (university level)
Edit: Actually, I just looked and it can also be used for the act of studying (for a test or class/doing homework)
For this and the one that translated to "they are drinking milk", is the sentence the same as "The girls read" or "They drink milk"? Asking because in the lesson previous, it said that things like "Hun drikker vand" was just "She drinks water" not "She is drinking water". Does the implication that the action is actively happening not change the sentence?
What I really can't understand is when the translation has the article and when not...
Why if I say that "Mænd spiser brod" the translation is "Men eat bread" but when I say "Pigerne læser" the translation is "THE girls read"? Which is the rule to understand when we must use the article and when not in the translation?