/u/XenophonTheAthenian [provides an answer that focuses more specifically on this clause within the context of the Twelve Tables] (https://www.reddit.com/r/AskHistorians/comments/cq9owp/in_the_twelve_tables_of_rome_women_are_legally/ewvmxzt/). My answer focuses more upon attitudes to female grieving in a later period and whether such practice could have influenced the introduction of said laws in the earlier period.

The grief of women was considered 'dangerous' to the Roman state. We do not really know much about this specific clause within the Twelve Tables due to a lack of evidence, but a later example following the battle of Cannae may shed light on a similar purpose.

In 216 BCE, following the defeat at Cannae, Livy paints a picture (one that is most likely exaggerated for literary purposes) of the women within Rome being struck with significant grief that posed a threat to the public order of the city (22.55.3-4):

But when, amid dangers at once so immense and so incalculable, they failed to think of even any tolerable plan of action, and were deafened with the cries and lamentations of the women, both the living and the dead —in the lack as yet of any announcement —being indiscriminately mourned in almost every house...

Following this, Q. Fabius Maximus Verrucosus proposed measures to try and keep public order (Livy. 22.55.6-8) - note the specific mention of women (matrons) again:

To discover and ascertain these facts was a task, he said, for active youths; what the Fathers themselves must do, since there were not magistrates enough, was this: quell the panic and confusion in the City; keep the matrons off the streets and compel them each to abide in her own home; restrain families from lamentation; procure silence throughout the City; see that bearers of any news were brought before the praetors —every man must wait at home for tidings that concerned himself; —and, besides this, post sentries at the gates, to keep anyone from leaving the City, and make the people rest all hope of safety on the safety of Rome and of its walls.

Now, such measures had no effect following the arrival of a letter from G. Terentius Varro which discussed the death of L. Aemilius Paulus and the destruction of his army that led to a disruption in the annual rites of Ceres, with Livy again identifying women as a cause of such disruption (22.56.4):

Announcement was then made from house to house of the losses they had each sustained, and the entire City was so filled with lamentation that the annual rite of Ceres was allowed to lapse, since it may not be performed by mourners, nor was there at that time a single matron who was not bereaved.

Now, Ceres was the god of argiculture and fertility and thus the disruption to the annual rites in the Roman mind may have been viewed as incredibly dangerous to an already fragile state if something was to happen to the grain production. Following this, the senate issued a senatus consultum that restricted mourning to a maximum of thirty days (Livy. 22.56.5). Both Livy and Valerius Maximus put the context of such action as being for religious matters (Livy. 22.56.5; Valerius Maximus. 1.1.15).

However, scholars, such as Hughes (Lament, Death and Destiny), suggest that the grief of women posed a threat to public order and political stability and such consultum as an attempt to counter-act the effects of such grief - Livy's account does focus upon the effect of female grief upon public order.

Such episode would suggest that within the Roman social mind that the grief of women was seen as something dangerous to the public order of the state and therefore considered as opposition to the stability imposed by the current political hierarchies. Thus, it was important to try and control the amount of mourning that could be done (it is not the only measure against women during the Second Punic War either).

 

Another idea is that the introduction of such law within the Twelve Tables was inspired by the laws of Solon in Athens. From Plutarch (Life of Solon. 21.4):

Laceration of the flesh by mourners, and the use of set lamentations, and the bewailing of any one at the funeral ceremonies of another, he forbade.

Here we see restrictions of women whilst mourning. Cicero (De Legibus. 2.59) informs us that such laws against women were directly copied from the laws of Solon. Literary tradition represents an embassy travelling to Athens in order to copy the laws which made up the basis of the Twelve Tables (Dionysius of Halicarnassus. 10.52.4).

 

As you can see, if we used the episode that happened during the Second Punic War. One could suggest that the restrictions upon grieving stem from threats to public order, the cessation of religious rites, the controlling of women who were seen as dangerous, or all factors.

It may have been the case that such law was simply copied from the laws of Solon also.

So /u/verrevert's answer does a good job of getting to a key point, namely that outward expressions of grief, particularly by women, have grave moral implications in the developed Roman state--whether that can be said to be true during the time of the decemvirs is a matter for another time. However, it doesn't really deal with what is sort of the crucial point here, namely why, in a legal code otherwise apparently unconcerned with moral and ethical problems (as opposed to the Augustan legal project), an ethical law appears. And that's basically because the prohibition on female grief is not really a moral one, although it certainly plugs into a larger fear of female grief in Roman society. It's a sumptuary law.

It's important to understand that the Twelve Tables are, like the Gortyn Law Code and the Hammurabic laws, presented and sorted generically. Unlike the Gortyn code and the Hammurabic text, which basically present larger laws and then dangle information on specific problems that might arise or court procedures that might be altered, the genera of the Twelve Tables are arranged according to the intent of the laws. All the laws of the Twelve Tables are intended to combat something, and they're arranged in groups according to what broadly they're combating. Just as importantly, the Twelve Tables are not a body of law generally, anymore than the Gortyn Code is or the later Justinian Codex, but are specifically a body of regulatory acts intended to structure and regulate social status. Therefore, no single law within the Twelve Tables can be properly understood without its context in the corpus as a whole.

And no wonder: most of the laws only survive as a single line or less, and many are totally nonsensical on the face of it. On the same tablet as the prohibition against female grief is a prohibition against collecting bones for reburial, which seems utterly absurd. Likewise, the prohibition on the same tablet against finishing off a funeral pyre with an ax. What on earth is going on here? Tablet 10, which contains the prohibition on female grief, contains a collection of laws regulating funerary ritual, and specifically the expense and spectacle of it. These are essentially sumptuary laws, and some of them quite literally are, such as the prohibition against using precious metals as grave goods and the allowing of winners of military honors to be buried with their ornaments. You mention professional mourners: this is exactly what this law is supposed to combat. And not just professional mourners, but the excessive shows of grief at funerals that are themselves displays of status. Funerals are large affairs, and the expense and spectacle of the event is an indication not only of individual status, but family status. Sumptuary laws directed at funerals are extremely common in the ancient world, practically universal in the Mediterranean. The most famous would probably be the Athenians' regulations--at Athens a complex series of laws prevented burials beyond a certain size, restricted the paths by which funerary processions could move to get to the burial plot, and eventually outlawed all funerary monuments except simple columellae bearing nothing but the dead's name. Now, as at Rome, the expression of excessive female grief was considered dangerous by the Athenians. There's a whole book on it, Dangerous Voices: Women's Laments and Greek Literature. But at it's core, the law is one of several prohibiting the excessive display of funerary processions and rituals. It's quite logical: just as burying the dead with a huge number of gold grave goods indicates that your family has the resources to throw that stuff away, a huge train with dozens or perhaps hundreds of female relatives and adherents sobbing and wailing loudly (naturally, the good Roman men should be expected to keep their peace, or something) is an indication for everyone of the prominence and massive power of such a family. If the procession passes someone who doesn't know who you are and how important you are, they certainly do by the time you've gone by. As such, this sort of thing functions as a regulation on aristocratic competition, which is something the Twelve Tables are extremely concerned with.