I heard of claims that the Greek City States were "proto-socialist". What are they referring to?

This is probably a reference to the fact that the Classical Athenian state, once it had completed its transition to a "radical" democracy, was heavily involved in income redistribution. Taxes were levied exclusively on the rich (as well as immigrants and merchants). The poor paid no tax and received pay from the state for a great number of duties. By the early 4th century BC, Athenian citizens were paid for jury service, attendance at the Assembly, military service, and rowing in the fleet. It is estimated that the Athenian state paid for the livelihood of something like 20,000 people, mostly at the expense of a leisure class that would have numbered a few thousand at most.

The principle behind this was not an ideological commitment to helping the poor and vulnerable, but rather an increasing commitment to the spirit of democracy. In an economy that was still partly based on subsistence, the poorest simply could not afford to take the time to participate in military campaigns or in political processes unless they were compensated in some way. The institution of jury pay and assembly pay therefore turned Athens from a theoretical democracy into a proper one. Once they were paid to attend, all adult male citizens could exercise their democratic rights and take part in political decision-making.

Meanwhile, income redistribution through pay for rowers was not ideological at all; it was simply a way to acquire good crews for potentially prolonged naval campaigns. Experience greatly improved the effectiveness of a trireme crew, and wages for experienced rowers and helmsmen were competitive; if Athens didn't pay enough, it might see its best rowers defect to Corinth or elsewhere. A large section of the Athenian poor made their living as rowers, which, as some ancient authors noted, made them likely to vote in favour of war. The process of taking money from the rich and giving it to the poor (through the state-imposed duty of the trierarchy, the responsibility to equip a trireme for one year) in this case was not a deliberate social programme, but merely a consequence of the way the Athenians manned and deployed their military forces.

Now, the Greeks idealised a life of leisure; they figured only a man free from want and from the requirements of daily labour was able to hone his skills as a politician, a philosopher, and athlete or a warrior. Therefore, for some, the income redistribution which the Athenian state was accidentally involved in did not go far enough. The fourth-century author Xenophon dreamt of a future Athens producing so much revenue through taxing trade and exploiting slaves in the silver mines that every free citizen could be paid a stipend that would suffice to make a living. This proto-Universal Basic Income/negative income tax was to make life better for all Athenians so that they could in turn devote themselves to the improvement of their community:

If the plans that I have put forward are carried out, I agree that, apart from the improvement in our financial position, we shall become a people more obedient, better disciplined, and more efficient in war. For the classes undergoing physical training will make more of an effort in the gymnasion when they receive their maintenance in full than they do under the superintendents of the torch races [the current leisure-class exercise instructors], and the classes on garrison duty in a border fort, or serving as peltasts, or patrolling the country will show greater alacrity in carrying out all these duties when the maintenance is duly supplied for the work done.

-- Xenophon, Ways and Means 4.51-52

Some of Xenophon's proposals seem to have been adopted after his death during the reforms of Euboulos, but these are very poorly attested. Once the Macedonians abolished the democracy in 322 BC, Athens largely stopped its institutionalised income redistribution, reverting to voluntary leisure-class philanthropy as was the standard in oligarchic states elsewhere in the Greek world.

I would be very hesitant to call any of this "proto-socialist" (just like I would not like to call Aristophanes' comedic scheme in the Assemblywomen or the property-sharing practices of Classical Sparta "proto-communist", as some have done). Socialism is a very particular political ideology that originates in 19th-century European socio-economic conditions and in Marx's philosophy of history. Neither its preconditions nor its goals were of any concern to the Greeks who created the laws of the Athenians.

It should also be noted that all Athenian policy took the availability of slave labour for granted, actively denied equal rights to immigrants, and basically tried its hardest to ignore the existence of women altogether. Its notion of "equality" (isonomia) would not pass muster in modern Socialist discourse.