When a comma is used after "no," it is as though the first "no" is a one-word sentence (No.) joined to another sentence (I don't have two pens.) English grammar allows commas to be used like this to join two short sentences when their message is logically related in some way. When a comma is used like this in an English sentence, the word "no" is called an "introductory element," as in the sentence "No, I don't have two pens."
In other words, in the sentence "No, no tengo dos plumas" the first "no" has the same meaning as the sentence "It is not correct" (No es correcta). The second "no" is a negation of the verb itself. In other words, "no tengo" = I don't have" and "tengo" = I have." In Spanish, using "no" before "tengo" is not optional if you mean to say "I don't have," but using the first "no," followed by a comma in both Spanish and English, is optional because it is used to emphasize, rather than to provide additional information. If you want to say "I don't have two pens," it is incorrect Spanish to use a comma before "tengo." To further illustrate how the first "no" works, consider the sentence "No, I have two pens" (No, tengo dos plumas). When would this sentence be used? Perhaps after the question "Do you have three pens?" (¿Tienes tres plumas?)
Note: This is not to be confused with the correct Spanish syntax rule of using more than one negative in a sentence like "No sé nada." This, translated word for word into English, is "I do not have nothing." However, in order to be correct English, this has to be translated as "I do not have anything."
In other words, while the Spanish grammar rule is always to negate the verb and follow it with a negative pronoun (nada, nadie, ninguno), the English grammar rule is that the verb is not negated when the object pronoun (for example: nothing, anything, something, nobody, anybody, somebody, some, no one, someone, anyone, none) is negated. That is why "No sé nada" is correctly translated as either "I don't know anything" (positive English direct object pronoun because verb is negated) or "I know nothing" (verb is not negated BECAUSE English direct object pronoun is negated).