Why do they use ¿? and ¡! in Spanish?

Answer 1

¿ and ? are used to indicate the beginning and end of a question.

¡ and ! indicate the beginning and end of an admiration or exclamation phrase. Such phrases are short, usually only one or two words. When speaking, you wanna say those louder. For example: ¡ayuda! ¡Sálvame! ¡Vamos ya! 

Answer 2

It's their way of indicating right at the beginning that it's a question or an exclamation. For example, in English one may not know early enough that it is a question and not just a statement. In Spanish, one is able to use the correct intonation easily because of the presence of the punctuation mark at the start of the sentence.