The reason for using a comma instead of a period depends on what purpose the dialogue is serving in the sentence.
"She's late again," mumbled Jackson.
In this case the dialogue, "she's late again," modifies the main verb, "mumbled."
"She's late again." mumbled Jackson.
This case, however, is improper grammar. "Mumbled Jackson" is a dependent clause, and needs the preceding "She's late again," to be a complete sentence. "She's late again" can be considered the object of the action; it is what Jackson is mumbling. The period needs to be replaced with a common, merely because the dialogue serves as a modifier.
Whether or not the dialogue needs a period generally depends on whether or not the narration can be a complete sentence without the dialogue a part of the sentence.
Jackson mumbled, "She's late again."
Jackson mumbled. "She's late again."
Both sentences, in this case, would be grammatically correct, though the meaning is slightly different.
Normally comma's are used to follow a speaker tag when introducing a quotation. The comma indicates that the quotations that follow are what the speaker is saying, mumbling, whispering, etc. The comma serves the same purpose when the dialogue precedes the speaker tag.
The latter example is grammatically correct, although replacing the comma with a period implies that Jackson is mumbling, perhaps because "she" is late, but doesn't imply that the quotations are what Jackson is mumbling.
Simply put, the commas clarify who is speaking and what that person is saying.