This quote from Macbeth is from Act I, Scene 3. This is at the beginning of the famous scene in which Macbeth and Banquo ~ fresh from battle ~ run into the 3 witches and hear the prophesy that Macbeth will become King.
The 3 witches are basically saying hello to each other, before Macbeth arrives, and one of the witches recounts that a sailor's wife refused to give her a chestnut to chew. The witch plans to follow after the woman's husband (but in a sieve I'll thither sail) and do horrible things to him (and like a rat . . . . I'll do).
It was said that witches "could sail in an egg shell . . . through and under the tempestuous seas" (Reginal Scott, Discovery of Witches, 1584) and that witches had gone to sea "each one in a riddle or cive" as part of a plot to drown the king at sea (Newes from Scotland, around 1591).
"Like a rat without a tail" refers to the belief that witches could assume the form of any animal they wanted, but that they would not have a tail. The basis for this was that no part of a woman's body corresponds to the long tail of an animal. Presumably the choice of a rat was because rats were so common aboard ships.
See The Plays of William Shakespeare with Notes by Johnson and Steevens, Vol VII. For some reason I can't link to this, but there is a digitized version of the book that you can find through Google.
'do and do and do' for me, describes the imbalance of MacBeth and his Lady. They have no contentment with where they are, and do one thing after another in an attempt to control their conditions. In the end, they bring ruin to themselves. The imbalance of excessive action and ambition, and a lack of contentment with what is.
So, yes, this is a witch talking, but her story is also a commentary on the main characters. It shows how bound up in action they are incapable of peace of mind. They are 'on the move' up the social ladder, and there is no rest for the wicked!