僕が何を怖がるっていうんだ？
It's a rhetorical question (修辞疑問文/反語表現). It's not really a normal question that asks for an answer/reply. It means/implies "What would I be afraid of? -- No, I would be afraid of nothing!"
Examples:
- 誰が知っているというのか。
"Who knows? (Nobody knows.)"- どうすれば忘れられるというのか。
"How could I forget? (I could never forget.)"- それがどうしたっていうんだ。*
"What would it matter? (It doesn't matter!)"- やつらが金以外のなにを欲しがるというんだ？*
"What do they care for but money? (They only care for money!)"
The latter two examples are taken from Weblio例文辞典.