So, I have this line like in the picture. I want to break it at those two pints, so there are like two ends. How can I achieve it?

I assume your "line" is not just a line (stroke) but a filled path (area)...
1. use the pencil tool (F6) to draw a line across your path through your two desired points
2. mark the line and your path
3. Menubar | Path | Divide Path
Also check this: (scroll down to Division path)
http://tavmjong.free.fr/INKSCAPE/MANUAL ... ining.html

Thanks!