Why do translations use words like "servants" to mean "slaves"?

No, the translation 'servant' is not because of political correctness.

Using 'servant' instead of 'slave' in, say, Romans 1:1 far predates political correctness as a modern movement. The Wycliffe-Purvey Version (1395) has it

"Paul, the servant of Jesus Christ"

The King James Version (1611), similarly, has it

"Paul, a servant of Jesus Christ"

Contrarily, more recent versions often do translate it as 'slave' or 'bond-servant' (which means 'slave').

The NET Bible

"From Paul, a slave of Christ Jesus"

You can contrast the King James with the New King James version, which has

"Paul, a bondservant of Jesus Christ"

Similarly, the NASB

"Paul, a bond-servant of Christ Jesus"

Whatever exactly is going on, it is clear that using 'servant' instead of 'slave' is not because of the recent trend called 'political correctness'.

One thing to keep in mind about why the Translators of the authorized version (KJV) of the Bible in English might have chosen to use the word "servant" instead of the word "slave" when translating the Greek and Hebrew terms, is that in the 17th century, there was no slavery in England. Serfdom, the nearest equivalent in England had been abandoned or abolished a hundred or more years earlier, and English listeners from the early 17th century would have had no understanding of the word "slave". Servants, though, were common, and may have been treated as poorly as slaves were, so this would have been a word that the early 17th century English listener would have understood.

The word servant used here is the Greek word doulos (δοῦλος, Strong's G1401). It can mean either a slave, servant, or bond-servant (similar to indentured servant). But in the Roman society of the time there was very little difference between them, since what we now think of as civil rights only applied to Roman citizens. Slave or not, a Roman master had every right to beat their servants simply because the servants were not Roman citizens. So I would argue that:

Translation philosophy

The translator's job is to pick the best word(s) to fit their translation philosophy. This could mean the most "literal" (i.e. technically accurate, ignoring connotations) term, or it could mean trying to most accurately capture the intent of the original author, even if that means changing the words. The first option is called "formal equivalence", while the later is called "functional equivalence". Most translators opt for a middle-ground approach where the words can be "changed", but only if there is a strong reason to do so. (This ignores that 1-to-1 correspondence is very rare. In reality, all translators must make such choices, regardless of philosophy.)

Slavery in Ancient Rome

It is true that δούλος most closely corresponds to "slave" from a technical standpoint. The problem is that slavery is typically inseparable from racism in the modern readers mind, at least in most of the English-speaking world (due to American slavery being the most familiar form). This would be a concept totally alien to the original New Testament writers. An estimated 95% of slaves in Italy were Italian born, for example. There is good reason to believe the concept of "race" didn't exist at all in Roman society.

While it is certainly true that conquered people by the Romans were sometimes sold into slavery, people could also sell themselves to pay a debt, and parents could sell their children into slavery.

Additionally, the modern reader associates slavery with forced, hard physical labor. This kind of slavery existed in Roman society too (e.g. mine workers), but it was not the only or even main kind of slavery. Almost any type of job you can think of, including cooks, doctors/nurses, housecleaners, private tutors, accountants, and so on, was a possible profession for a slave. (At least 55 distinct slave jobs are known.) Slaves with such skills were highly valued; cooks in particular enjoyed a high social status because of their value in entertaining guests of wealthy patrons.

Roman slaves were allowed to earn a wage (page 147) and could, in principle, buy their own freedom (although it was very hard to do). Significant social mobility was not unheard of for skilled slaves. Slaves that were freed by their masters choice often chose to keep the same job. Thus, there was little practical difference between a technically free servant and a slave. The Latin word servus (from which we get servant) could refer to either. (The word sclavus, from which we get slave, was not invented until very late in Latin history, well after the Biblical period.)

Implication for translation

What this all means is the that the first century conception of "slave" and the twenty-first conception are very different. A translator is thus left with a difficult choice - he can either translate as "slave" and possibly add racist and/or violent overtones to a text where there was none originally, or he can translate as "servant" and lose the lack of freedom vibe.

The normal practice in Latin texts (where servus can mean either slave or servant) is to translate the word as servant in most situations, except when the immediate context only fits a "slave" translation. (E.g. when talking about the activities of the cook, the cook is a servant; when talking about the rules regarding buying and selling cooks, the cook is a slave.) This is done for reasons of accuracy, not "political correctness", and applies to texts of all types, including those that have zero to do with Christianity.

Even though Greek does have a separate word for servant, it is still a legitimate translation practice to translate δούλος as "servant" when the context does not demand a "slave" reading. (Only a hyper-literal translation philosophy would demand otherwise.) This is not done to "hide" anything from the reader, but rather is done because the idea of a "slave" in the culture of the writer more closely matches the idea of a "servant" in the mind of the modern reader in those passages.

Luke 12:45-48

Turning to the passage that prompted the question, it is seems likely that the δούλος described in this passage are household slaves. (That is, that they do jobs that a modern reader would associate with servants.) I say this because a master leaving the household for an extended period is a more natural occurrence than a master leaving mine workers, for example, unattended for an extended period. Additionally the description of the δούλος eating and drinking excessively implies he may be in charge of such provisions.

Thus, the "servant" of the NIV (also ESV) is justifiable under a more functional equivalence based approach. "Slave", which is used by NRSV and NASB, is also justifiable, under a more formal equivalence based approach. The meaning of this passage does not in any way change based on the translation of δούλος, so any suggestion that the translation choice is made for theological reasons is off-base.

Addendum

After writing this answer, it came to my attention that the ESV actually explicitly states how they translate δούλος:

Third, a particular difficulty is presented when words in biblical Hebrew and Greek refer to ancient practices and institutions that do not correspond directly to those in the modern world. Such is the case in the translation of ‘ebed (Hebrew) and doulos (Greek), terms which are often rendered “slave.” These terms, however, actually cover a range of relationships that require a range of renderings—either “slave,” “bondservant,” or “servant”—depending on the context. Further, the word “slave” currently carries associations with the often brutal and dehumanizing institution of slavery in nineteenth-century America. For this reason, the ESV translation of the words ‘ebed and doulos has been undertaken with particular attention to their meaning in each specific context... In New Testament times, a doulos is often best described as a “bondservant”—that is, as someone bound to serve his master for a specific (usually lengthy) period of time, but also as someone who might nevertheless own property, achieve social advancement, and even be released or purchase his freedom... Where absolute ownership by a master is in view (as in Romans 6), “slave” is used; where a more limited form of servitude is in view, “bondservant” is used (as in 1 Corinthians 7:21–24); where the context indicates a wide range of freedom (as in John 4:51), “servant” is preferred.

A few thoughts on the King James

The above answer only applies to recent translations. The 1611 King James Version obvious did not make its decisions based on the later American slavery. Ultimately, it speculation to define their motives for mostly avoiding "slave" (then spelled "slaue"), but I see several possibilities:

• They did so for "politically correct" reasons. This seems unlikely to me since slavery was legal and probably "socially acceptable" in 1611.
• They did so for reasons of personal conviction against slavery.
• They were influenced by the use of servus in the Latin Vulgate.
• "Slave" was a relatively rare word, and as such was avoided. I think this unlikely - see Google N-grams.
• "Slave" could be confused with other uses of the term in contemporary English. In support of this hypothesis, see the following Oxford English Dictionary definitions that are now archaic, but were in use in 1611:
  • Used as a term of contempt:

    Where is that Slaue Which told me they had beate you to your Trenches? (Shakespeare, Coriolanus, I. vii. 39)

  • In less serious use: Rascal; fellow.

    Oh Slaues, I can tell you Newes, News you Rascals. (Shakespeare, Coriolanus, IV. v. 175)

I'll leave it to the reader to draw their own conclusions about the KJV.