"There's a part of me that feels ~"
"there's" = "there is" 「〜がいる」
"a part of me" 「自分の部分」・「どこかの自分」
"that feels ~" 「〜を感じる」
"a part of me that feels ~" 「〜をかじる自分」
"There's something that makes me think ~"
"there's something" 「何かがいる」
"that makes me think ~" 「〜で思いさせる」・「〜を感じさせる」
"think" の代わりに "feel" もつかえます。