See this all the time. Doing it right now. Leaving out the "I am" piece of the sentence. Does this have a name?
This is an example of pro drop (short for pronoun dropping).
Some languages, like Spanish, pretty much mandate it. Such languages are called pro-drop languages.
In English, most grammarians would probably consider it ungrammatical, but it's pretty standard in informal speech.
Regarding its occurrence in English, Wikipedia writes:
English is considered a non-pro-drop language. Nonetheless, subject pronouns are almost always dropped in imperative sentences (e.g., Come here). In informal speech, pronouns may sometimes be dropped in other types of sentences...
- [Have you] ever been there?
- [I'm] going to the shops. [Do you] want to come?
- Seen on signs: [I am/We are] out to lunch; [I/we will be] back at 1:00 [P.M].
The second and third examples of this passage are precisely the kind of sentences you ask about. These sentences can be called pro-dropped sentences.
Note, however, that pro dropping can occur for any pronominal subject, not just for "I". For example, consider the following dialogue:
A: Where's Ingmar?
B: Went home.
Here, the pronoun "he" is being dropped.
As a final note, these kinds of pro-dropped sentences fall under the more general umbrella of ellipsis:
the omission from speech or writing of a word or words that are superfluous or able to be understood from contextual clues.