웃겨 is a conjugation to the 아/어/여 ending, which is both a regular ending used in the middle of a sentence (사람을 웃겨 돈을 버는 직업) and a sentence ending ("이 코메디 프로(는) 너무 웃겨."). The exact form of the -아/어/여 ending of verbs and adjectives is determined by the vowel of the syllable that comes before it (a phonetic phenomenon called 모음 조화 = vowel harmonization) 1. after ㅗ,ㅏ(except 하다) => 아 (돌다 -> 돌아, 보다 -> 보아 -> 봐). 2. after all others => 어 (밀다 -> 밀어, 먹다 -> 먹어). 3.. after 하(다) => 여 (하다 -> 하여 -> (usually) 해). So 웃기다 conjugates to 웃기어 since its stem ends with 기(vowel ㅣ), and then typically contracts to 웃겨 (ㅣ+ㅓ -> ㅕ).