〜てみせる

The base meaning is actually performing an action to show it to someone (e.g., 手本を示す). From this, in the form 「必ず〜してみせる」, comes a usage expressing strong determination or a declaration, vowing to those around one that one will accomplish it by one's own power even if difficult. In this determination usage, it is used as 「合格してみせる」「勝ってみせる」. Whether it is an actual demonstration or determination is decided by context.