Since 就 is a construct that has many meanings, I'm only going to concentrate on your sentence.
In your sentence it will give emphasis that it is really early. You can also write:
我明天早上很早走 but this lacks the emphasis 我明天早上很早就走 has. And the latter construction will be used much more than the former.
Some similar constructions:
- 马上就来: to come right down
- 一会儿就好: I'm ready in a moment
It is used for actions (after the 就). You have the structure A就B and B will be done after A is done.
就 can be used to say that one thing comes immediately or very soon after the other. For example 我明天吃了早饭就走: Tomorrow, I will immediately leave after having eaten breakfast.
In this type of sentence you normally have a verb before the 就 and when after the 就. Although there is no verb in 我明天早上很早就走, it is actually the same kind of structure and you can understand it the same. Immediately after "早上很早" you'll leave.
The opposite of this 就 is 才 (another word with many meanings). There you emphasize that something happened really or rather late. For example 我十一点才起床了： I only got out of bed at 11.