She was in a hurry (she didn't have much time), and she wrote a few lines (of text) for a person (a man). She wrote them on the back of an envelop. If it doesn't help you, please say what you don't understand exactly.

The passage implies that she felt the need to respond to him quickly, and not having paper, improvised and used whatever she had on hand-in this case, an envelope.

"envelope"

she used the back of the envelope to write several lines of the message quickly

The back of an envelope is blank, and the front of an envelope doesn't need to be saved. People may write on the back of an envelope because they're in a hurry and can't find a clean sheet of paper. Or they may do it because they are frugal and don't want to waste a piece of paper on a scribbled note that will be read and immediately thrown out. I literally do leave notes for my wife on the backs of envelopes (and vice versa). There is saying, "a back-of-the-envelope calculation," meaning a piece of arithmetic that's too complicated to do in your head, but just a quick, rough calculation. Another similar saying--which also really happens--is "to make a sketch on a napkin." The real-life scene is that you are at a restaurant with friends, and suddenly you need to sketch something in order to explain it, so you use a (paper!) napkin to draw on.