Unlike English, Chinese sentences can sometimes go without a verb. In the sentence 我很好, the adjectival phrase 很好 serves as the predicate for the sentence. No verb "to be" is needed as in English (I /am/ very well). This is generally true of a sentence that has an /adjective/ as its predicate, even in negative sentences.
e.g. 他們很忙, 我們不忙
They are very busy, we are not.
Notice that in other types of sentences, however, verb 是 should still be retained:
他是我朋友
He is my friend.
我不是她的哥哥
I am not her elder brother.
Study the following two dialogues:
A: 你媽媽是大夫嗎?
Is your mother a doctor?
B: 是, 她是大夫.
Yes, she is a doctor.
A: 你媽媽忙嗎?
Is your mother busy?
B: 忙, 她很忙.
Yes, she is very busy.
To respond positively to a yes/no question, we often repeat the verb -- in the above case 是 and 忙 respectively -- as an equivalent to the English "yes" before making the statement itself. Note therefore that the response is not always 是.