In act four, scene one, Macbeth visits the Three Witches for a second time in order to learn more about his future and whether or not he will be able to cement his legacy as King of Scotland. The Three Witches proceed to conjure several apparitions, which purposely mislead Macbeth and influence him to be overconfident. The first apparition warns Macbeth to beware of Macduff without giving him any further context. The second apparition encourages Macbeth to be "bloody, bold, and resolute" because no man born of a woman can harm him. Macbeth accepts this prophecy literally and reasons that even Macduff cannot harm him since every man is born of a woman. The third apparition also misleads Macbeth into becoming overconfident by telling him that he shall not be defeated until Birnam wood travels to Dunsinane hill, which is something Macbeth believes is impossible.

Despite the seemingly favorable prophecies, the fourth vision disturbs Macbeth as he witnesses a parade of eight kings and recognizes them as Banquo's descendants, which confirms Banquo's initial prophecy. After the witches disappear, Lennox arrives and informs Macbeth that Macduff has fled to England. Macbeth responds by saying,

The flighty purpose never is o'ertook

Unless the deed go with it. From this moment

The very firstlings of my heart shall be

The firstlings of my hand (Shakespeare, 4.1.161-164).

Macbeth is saying that unless a person does something the second he thinks of it, he will never get a chance to do it. Therefore, he will begin acting on his impulses by turning his thoughts into actions the second he thinks of them. Macbeth then instructs his agents to slaughter Macduff's entire family. Macbeth's quote emphasizes his change in his character as he becomes a completely impetuous, bloodthirsty tyrant who acts upon his impulses and is willing to kill all of his political enemies.

The line quoted above is found in act 4, scene 1 of Shakespeare's Macbeth and is spoken by the titular character himself in the speech that closes out the scene.

For context, Macbeth has just visited again with the weird sisters (the three witches) to receive another prophecy of his future. The witches tell him that he will never be defeated until Birnam Wood, a forest in Scotland, advances on the castle Dunsinane. This would seem to be an impossible occurrence and therefore Macbeth is heartened, but he is still made anxious by the encounter and worried about maintaining his position.

After the witches leave, Lennox enters with news that Macduff has fled to England, and it is then that Macbeth delivers the speech that contains the line you have quoted. Macbeth is upset that Macduff escaped before he could enact his plans. In the line you quote above he makes a promise to himself that in the future when an idea occurs to him (in his heart) he will act on it without wasting any time (with his hands).

This quote is found at the end of Act IV, scene one, as the play nears its climax. Falsely reassured of victory by the witches, Macbeth reacts in frustration to the news that Macduff has fled to England and escaped his grasp. Macbeth then declares that he won't let anything like that happen again. From now on, he will act immediately on his desires, without hesitation. This is what he means when he says the "firstlings of my heart shall be the firstlings of my hand."

There's irony in this speech: what has Macbeth done throughout the play but act on his desires, with ever worsening results? Arguably, he's learned nothing. The wisest course he tried to take was to reconsider his desire to murder Duncan. Acting on that desire led him down a path toward greater and greater disaster. Now, however, he decides he is going to double down and plunge into more murder. 

One also has to question his bold contention that he will see "no more sights," such as the ghost of Banquo. All along, he and Lady Macbeth have been declaring their toughness, but conscience continually overtakes them despite their bravado. Why should Macbeth suddenly attain the hardness he hasn't thus far achieved? An audience wouldn't be wrong to take this speech as more evidence that Macbeth is dissolving and is an unfit ruler.

The passage about which you are asking is found at the very end of ACT IV scene 1. The Weird Sisters have just shown Macbeth a series of visions which make him feel very confident that he cannot be defeated yet also very anxious. After the sisters leave, Lennox arrives, and tells Macbeth that Macduff has fled to England.

Macbeth vows to let nothing stand in the way of his dreams/plans in the following lines:

Time, thou anticipat'st my dread exploits:
The flighty purpose never is o'ertook
Unless the deed go with it: from this moment
The very firstlings of my heart shall be
The firstlings of my hand. And even now,
To crown my thoughts with acts, be it thought and done:
The castle of Macduff I will surprise;
Seize upon Fife; give to the edge o' the sword
His wife, his babes, and all unfortunate souls
That trace him in his line. No boasting like a fool;
This deed I'll do before this purpose cool:
But no more sights!--Where are these gentlemen?
Come, bring me where they are.

What Macbeth is saying is that whatever his heart first tells him to do, no matter how heinous or bloody, his hands will follow through with the action--including the murder of Macduff's family.