내게 is a contraction of 나에게, 나(I) + 에게(to), so it means "to me" in the simple case.
In conversation, 나한테 is often used in place of 내게.
- 그거 내게 줄래? = 그거 나한테 줄래?: Will you give that to me?
- 내게 말해 봐 = 나한테 말해 봐: Why don't you tell me (about that).
- 어머니가 나에게[나한테] 쓰신 편지: The letter my mother wrote me.
내게/나에게 has more uses than the above simple ones.
To understand it fully, you need to know the usages of the particle(조사) "에게".
~에게 has three major uses according to the dictionary:
1. to (as in "something has an effect TO", or "belongs/pertains TO")
- 그에게 무슨 일이 생겼나?: What happened to him?
- 내게(는) 친구가 많다: I have many friends (There are many friends to/for me).
2. to (a direct destination/target of an action/effect)
- 친구들에게 이 사실을 알렸다: I told my friends about this.
- 돼지에게 먹이를 주었다: I gave forage to the pig.
3. from, with, by (a source of an action/effect)
- 이 아이는 친구들에게 놀림을 받는다: This child is mocked by his friends.
- 나는 개에게 물렸다: I was bitten by a dog.
As you can see, ~에게 always involves a person or animal, and can be replaced by 한테 most of the time.
#3 usage may be hard to understand, as it seems to indicate the opposite of #2 (to vs from).
But both are common usages. In most cases the verb make the direction of effect clear, so it is not confusing.