In this context, to say she "made it through" to the semi-finals means that she has not been eliminated from the contest. This phrase can be used in the context of something with multiple stages. In the Olympics, for example, let's say the soccer team from your country "made it through" ten games, meaning your team played ten matches and won all ten. To "make it through" can refer to passing tests, surviving something (like surgery, for example), or completing something challenging. I hope this helps!