Roman slaves were legal unpeople, having suffered what the scholar Orlando Patterson calls "social death." This social death meant that family bonds of slaves were not recognized; legally slaves were not considered to have parents, siblings, children, and they were not able to contract marriages. Never mind them having standing to bring lawsuits-- slaves were not even considered trustworthy enough to testify in court except under the threat of torture. Furthermore, it was a fundamental principle of Roman ideology that the head of the household (paterfamilias) had patria potestas, complete control over the people in his household, up to and including the power of life and death. This all would seem to make it a legal impossibility for an enslaver to murder a slave, or for a slave to sue an enslaver.

And yet: both of Wikipedia's claims are basically true. How is this possible? Well, for a Roman slave to sue their master, they just had to hug the right statue, doncha know.

Let's start with patria potestas. Note that it is a principle of Roman ideology and not of Roman law. It's common for a state to have sunlight between its ideology and its reality. Consider how a country might have "All men are created equal" as a fundamental ideology and yet have quite a lot of inequality in practice. Early Roman lore is full of stories of fathers taking it on themselves to kill their own sons and daughters over matters of honor, and they are celebrated as tragic heroes for making this difficult choice. But these stories are suspiciously concentrated in the murky mytho-historic days of the Early Republic. If patria potestas had ever been an absolute right (and the Romans certainly fantasized that it once had), those days were long over by the Early Imperial period. A father arbitrarily killing his son would not be seen as the lawful exercise of his patria potestas, it would be seen as murder. The Late Republican politician Catiline was accused of having poisoned his own son to persuade a woman who didn't want a stepson to marry him. In Cicero's oration against Catiline, he pointedly reminds his listeners of this accusation, then cuts himself short and says that he will "pass that over, and willingly allow it to be buried in silence, that so horrible a crime may not be seen to have existed in this city." Whether or not Catiline had actually done this (it's the sort of soap-opera accusation Romans loved to lob wildly, so feel free to doubt it) it was clearly understood as a crime, not as a legitimate exercise of patria potestas. (This bit is drawing on Emma Southon’s A Fatal Thing Happened on the Way to the Forum.) By the Early Empire, wives were not under their husband's patria potestas at all anymore: the earlier custom of "manus marriage" where they were transferred from the power of their father to the power of their husband had been almost completely replaced by "free marriages" where the wife remained under the nominal patria potestas of their father, maintaining their own property that travelled with them as they married and divorced and remarried.

This weakening of patria potestas was already old news by the Early Empire. But it's under the Empire, not the Republic, that we start seeing new laws protecting slaves appear. The article on Wikipedia you're referring to suggests that this is due to the "influence among the educated elite of the Stoics, whose egalitarian views of humanity extended to slaves." This might be one factor but I'm skeptical that it's the main one. The Stoics tended to view slavery in the context of personal morality and self-control: their emphasis was on teaching enslavers to be kind towards their slaves, not on any kind of structural changes to slavery. And the early emperors were pretty hot and cold on Stoicism anyway. Emperor Claudius exiled the Stoic philosopher Seneca as soon as he took power. There was a brief peak of Stoic power in the 50s when Seneca was called back to Rome to serve as a guiding influence on young Nero. But Nero started to chill on Stoicism in general and Seneca in particular in the 60s. The infamous Praetorian prefect Tigellinus persuaded Nero to exile one philosopher by (per Tacitus) saying he "had the arrogance of the Stoics, which makes men restless, and eager for a busy life" (with the implication, in context, that Stoicism leads to intrigue). Nero forced Seneca into retirement in 62, and then fatally purged him and other Stoics after the botched Pisonian Conspiracy in 65. Stoicism did not become politically safe until after the fall of the Flavian dynasty in 96 CE, by which point legislation controlling abuse of slaves was already a Thing. In my opinion, the trend towards legislation on slavery in the Early Empire should be seen in the context of the emperor becoming the pater patriae-- the "Father of the Country," every Roman's patron and every Roman's paterfamilias. It is in continuity with Augustus's morality laws encouraging Romans to marry and have children, and even his laws limiting slave manumissions. This might seem odd at first glance: limits on slave manumissions are "anti-slave" and laws like e.g. Claudius's rule granting freedom to slaves abandoned on an island by masters who didn't feel like paying their medical bills are "pro-slave." However, the common thread is that they're both asserting that it is the emperor's business to govern the morality of the Roman household. This is less Stoic egalitarianism and more Augustan paternalism.

So the Roman paterfamilias did not have absolute patria potestas, and the Roman emperors were getting more involved in the household. Even with those factors enabling emperors to give slaves legal rights, why would they bother to? I've indicated in two past answers (1, 2) that I don't believe there was any real opposition to slavery as an institution in Rome. But just because Romans didn't oppose slavery doesn't mean that they didn't care what happened to slaves. Unlike in American slavery, race did not create a hard social barrier between the free and the enslaved. Romans manumitted their slaves at a rate that boggled the minds of their non-Roman contemporaries, and manumission came with either automatic Roman citizenship or clear pathways to it. This meant it was very common for freeborn Roman citizens (and even the odd senator) to count freedmen and slaves among their family members or recent ancestors. Social interaction with slaves was an everyday fact of life for all Romans. Slaves were not just laborers but also clerks, merchants, property managers, doctors, educators, childcare providers, professional athletes, celebrity actors-- there were few jobs that were reserved only for the free, or only for the enslaved. If you were an enslaver, slaves would be an integral part of your household. It was very normal for letters between Roman enslavers to inquire how their wife, children, and slaves were doing. This social intimacy between Romans and slaves sometimes led to marriage. Roman law is quite detailed about the various and sundry consequences of free Romans marrying or cohabiting with freedmen and slaves, and while I won't drag you through the details the upshot is this: 1. men could manumit and marry whoever they wanted 2. freeborn women could manumit and marry freedmen but it was seen as icky and elite women were barred from doing it 3. freeborn women were not supposed to cohabit with slaves, and could under some circumstances could be enslaved themselves if they did, but it was apparently common enough to need detailed rules addressing a variety of circumstances. (If you want the details, you can check out Evans-Grubb's "Marriage More Shameful Than Adultery": Slave-Mistress Relationships, "Mixed Marriages", and Late Roman Law.)

None of this protected Roman slaves from physical or sexual abuse. Roman undertakers often ran a side business hiring themselves out as torturers to punish slaves; in this earlier answer I discuss why being assigned to mill flour at a bakery was a particularly cruel punishment. Romans generally disapproved of physically abusing slaves without "a good reason" but in the way you might disapprove of a parent who feeds their kid an all-pizza diet: it reflects badly on their character but is fundamentally an issue internal to their household. Romans did not generally disapprove of sexual coercion of slaves; it was taken for granted that all slaves would be sexually available to their enslavers. Slavery is inherently abusive, and the Romans tortured, raped, and exploited their slaves. When people say (as they occasionally do) that Rome had a more humane system of slavery, please understand that this is rather like praising Pol Pot for having a lower body count than Hitler. BUT: almost all free Romans would have had slaves in their life that they knew, liked, and sympathized with.