Mater ungues infantis secat, et serva manicas ad focum suspendit, ut citius siccentur.

Because infantis means of the baby/child and shows possession: the baby's nails.

• ungues infantis = the nails of the baby
• Latin often uses the genitive where English uses 's or of

So infantis depends on ungues, not on secat.

Because the sentence is talking about nails in general, not just one nail.

• unguis = nail, claw
• ungues = nails

Latin, like English, often uses the plural here because a person normally has more than one nail being cut.

Ungues is accusative plural, because it is the direct object of secat.

• mater = the subject
• secat = cuts
• ungues = what she cuts

So the structure is:

• Mater = mother
• ungues infantis = the baby's nails
• secat = cuts

Because manicas is the direct object of suspendit.

• serva = the slave-girl/maidservant
• suspendit = hangs up
• manicas = sleeves

So the servant is hanging up the sleeves, and therefore manicas must be in the accusative.

Ad focum literally means to/toward the hearth or near the fire.

• ad
  • accusative often means movement to or toward
• focus means hearth, fireplace, fire

In context, suspendit ad focum means she hangs the sleeves up by the fire / near the hearth so that they dry.

English might prefer by the fire, but Latin often uses ad in this kind of practical expression.

Because it is in an ut-clause of purpose.

• ut
  • subjunctive often means in order that or so that
• ut citius siccentur = so that they may dry more quickly

This is one of the most common uses of the subjunctive in Latin: expressing purpose.

Because its subject is manicas.

The sleeves are the things that are supposed to dry, so the verb must be plural:

• manicas = plural
• siccentur = may be dried / may dry, plural

Even though manicas is accusative in the main clause, it is understood as the subject of the subordinate clause idea: so that the sleeves may dry more quickly.

Citius is the comparative adverb from cito, meaning more quickly.

• cito = quickly
• citius = more quickly

Latin often uses cito, citius, citissime for speed.
So ut citius siccentur means so that they dry more quickly.

A learner might expect celerius from celer, and that would also relate to speed, but cito/citius is very common and idiomatic here.

Latin word order is much freer because the endings show each word's function.

So instead of depending heavily on position, Latin uses case endings and verb endings:

• mater is nominative, so it is the subject
• ungues and manicas are accusative, so they are objects
• infantis is genitive, so it shows possession
• siccentur is clearly a verb in the subordinate clause

That means Latin can arrange words for style, emphasis, or rhythm.
This sentence is still fairly straightforward, but it does not need to follow a rigid English-style order.

Yes. Serva literally means female slave or maidservant.

In many beginner Latin passages, it is often translated more gently as servant or maid, depending on context. Grammatically, though, it is simply:

• serva = nominative singular feminine
• subject of suspendit

So the important grammar point is that serva is a second subject, coordinated with mater by et:

• Mater ... secat
• et serva ... suspendit

It has two main clauses joined by et, followed by a purpose clause.

1. Mater ungues infantis secat
   = main clause

2. et serva manicas ad focum suspendit
   = second main clause

3. ut citius siccentur
   = purpose clause modifying the second action

So the logic is:

• the mother cuts the baby's nails,
• and the servant hangs the sleeves by the fire,
• so that they dry more quickly.

Formally, siccentur is passive in form, from sicco, siccare, to dry.

So very literally it can mean may be dried. But in English, with clothing or sleeves, we often translate more naturally as dry:

• literal-ish: so that they may be dried more quickly
• natural English: so that they dry more quickly

So the passive form is normal Latin, even if English often prefers a more natural active-sounding translation.