Strange as it might seem, these gestures were not made up for the show, but seem to have been a real feature of Roman oratory. The Romans recognised, as we do, that body language says as much about a person’s thoughts and intentions as the words they say. It was therefore believed to be incredibly important for an orator to hold and move his body in ways that would aid him in convincing his audience, whether in the senate or law courts (or both!). Cicero himself wrote a guide to oratory (De Oratore), and it includes this short section on gestures:

But all these emotions must be accompanied by gesture—not this stagy gesture reproducing the words but one conveying the general situation and idea not by mimicry but by hints, with this vigorous manly throwing out of the chest, borrowed not from the stage and the theatrical profession but from the parade ground or even from wrestling; but the movements of the hand must be less rapid, following the words and not eliciting them with the fingers; the arm thrown out rather forward, like an elocutionary missile; a stamp of the foot in beginning or ending emphatic passages.

Although it was written a little later (1st century AD), Quintilian’s guide to oratory goes into much more detail. I’ve quoted a small section of book 11.3, which deals with delivery and includes advice on vocal style as well as the gestures and movement of all parts of the body (you can read the full section here).

(1) The commonest Gesture consists of bending the middle finger against the thumb and extending the other three. This is useful (a) in the Prooemium, the hand being moved slightly forwards with a gentle movement to either side, and the head and shoulders gradually following the direction of the hand. It is also (b) an assured Gesture in the Narrative, but then it should extend somewhat further forward, and (c) a Gesture of energy and insistence in reproaches and refutations, because here the hand is extended further and more freely. On the other hand, it is wrong for this movement to be directed (as it often is) sideways, as though aiming at the left shoulder; even worse, some people hold the arm out across the body and make their statement with their elbow. (2) The two middle fingers can also be turned under the thumb; this is an even more insistent Gesture than the last, and is not suitable for Prooemium or Narrative. (3) When three fingers are doubled under the thumb, the finger which Cicero says Crassus used so well is extended. This finger is important in reproach and in pointing things out (which is why it has its name). Turned slightly downwards, with the whole hand raised and turned towards the shoulder, it expresses strong statement; pointed down towards the ground, facing downwards, as it were, it insists on a point. Sometimes also it indicates number. (4) Again, if the top joint of this finger is gripped lightly on both sides, and the two last fingers are slightly curved (the little finger rather less so), the Gesture is appropriate in arguing a point. (5) Those who prefer to hold the middle joint, and contract the two last fingers more, to match the lower position of the thumb and the middle finger, give the impression of a sharper level of debate. (6) A Gesture particularly well adapted to an expression of modesty consists in bringing the thumb and the first three fingers gently together to a point, and moving the hand towards the body in the region of the mouth or chest, then letting it fall, palm downwards and slightly brought forward. This, I fancy, was the Gesture with which Demosthenes began the timid and subdued Prooemium of his defence of Ctesiphon; this is surely how Cicero held his hand when he said “If, members of the jury, there is any talent in me—and I know how little it is.” (7) The hand may also be drawn towards the body, with the fingers pointing down a little more freely, and then opened more widely to face the opposite way, so that it seems to be somehow delivering our actual words. (8) Sometimes, we separate the first two fingers from the others, not however inserting the thumb between them, but with the fourth and little fingers turned slightly inwards, and the other two also not fully extended. (9) Sometimes again the two last fingers press on the palm around the base of the thumb, while the thumb itself is pressed against the middle joints of the two first fingers. (10) Sometimes the fourth finger is bent back to the side. (11) Sometimes, by relaxing rather than extending all four fingers, and letting the thumb incline inwards, we produce a hand which is useful for pointing to one side or marking breaks in what we are saying; the hand moves palm-upwards towards the left, palm-down towards the right.

The big question for us in the modern world is why on earth did the Romans go to such specific and frankly bizarre lengths to control the movements and comportment of their bodies? There has been a fair amount of scholarship recently that is getting into these questions, and I think it's doing quite a lot to give us new and exciting understandings of elite Roman society and culture.

These texts clearly show a society that was obsessed with presentation and appearance, and it seems that elite males must have operated under the constant scrutiny of their peers. From Quintilian's guidance we can see that the slightest bend of a single finger, or the tilt of the head could have a serious positive or negative impact on the success of a speech. It follows that, as performances in the political and legal debates were essential elements in an elite Roman's career, getting these gestures right were vital for advancing their position in their cut-throat, competitive world. Furthermore, as a successful public career was a central aspect of virtuous, masculine identity in the Roman world, the correct use of gesture and bodily movements would seem to be at the core of what it meant to be a Roman man.

There's very good evidence for this last point - the peer group of Roman elite men were ruthless in identifying and criticising movements and gestures that could be interpreted as unmanly or feminine. In one of his letters (52.12-13), Seneca says the following:

If you mark them carefully, all acts are always significant, and you can gauge character by even the most trifling signs. The lecherous man is revealed by his gait, by a movement of the hand, sometimes by a single answer, by his touching his head with a finger, by the shifting of his eye. The wicked man is shown up by his laugh; the madman by his face and general appearance. These qualities become known by certain marks; but you can tell the character of every man when you see how he gives and receives praise.

Elsewhere in On Oration (5.9.14), Quintilian says this:

Depilation, a voluptuous gait, or womanish attire may be regarded as indications of effeminacy and unmanliness by anyone who thinks that such symptoms are the result of an immoral character, just as blood is the result of a wound: for anything, that springs from the matter under investigation and comes to our notice, may properly be called an indication.

According to his biographers, the care that Caesar took over his appearance - particularly his clothing and hair, but also the way he moved his hands and head - caused confusion among his peers as to his political intentions: either is suggested he was very dangerous, or totally harmless.

Suetonius, Life of Caesar, 45.

He was somewhat over the top in the care of his person, being always carefully trimmed and shaved. Some have accused him of even having superfluous hair plucked out. While his baldness was a disfigurement which troubled him greatly, since he found that it was often the subject of the gibes of his enemies. Because of it he used to comb forward his scanty locks from the crown of his head, and of all the honours voted him by the senate and people there was none which he received or made use of more gladly than the privilege of wearing a laurel wreath at all times. They say, too, that he was remarkable in his dress; that he wore a senator's tunic with fringed sleeves reaching to the wrist, and always had a belt over it, though rather a loose one; and this, they say, was the occasion of Sulla's quip, when he often warned the nobles to beware of the boy with loose clothes.

Plutarch, Life of Caesar, 4.8-9.

At all events, the man who is thought to have been the first to see beneath the surface of Caesar's public policy and to fear it, as one might fear the smiling surface of the sea, and who comprehended the powerful character hidden beneath his kindly and cheerful exterior, namely Cicero, said that in most of Caesar's political plans and projects he saw a tyrannical purpose; "On the other hand," said he, "when I look at his hair, which is arranged with so much nicety, and see him scratching his head with one finger, I cannot think that this man would ever conceive of so great a crime as the overthrow of the Roman constitution."

I should probably sum up here. Yes, the gestures depicted in HBO/BBC Rome are historical, in that they are found in the guides to oratory written by Cicero and Quintilian. They mattered because the movements and gestures of elite Roman males,especially while making speeches, were seen as demonstrative of their personal character and virtue. Moving and gesturing in the right way could win you a case or a senatorial debate. Doing it wrong could do fatal damage to your public reputation.

This, more general, thread on the use of gestures in the show may be useful to you.

Especially /u/XenophonTheAthenian 's answer at the top.