Na prvom slajdu projekt je sažeto objašnjen, a na drugom su primjeri.

Because na here means on / on the slide, which expresses location, so Croatian uses the locative case.

• slajd = slide
• na slajdu = on the slide
• na prvom slajdu = on the first slide
• na drugom = on the second one

In the second part, slajdu is omitted because it is understood from the first part. So:

• a na drugom su primjeri

really means:

• a na drugom slajdu su primjeri

This kind of omission is very common and natural.

Because the adjective has to match the case of the noun.

Here, slajdu is in the locative singular, so the ordinal numbers/adjectives must also be in the locative singular:

• prvi slajd = the first slide (nominative)

• na prvom slajdu = on the first slide (locative)

• drugi slajd = the second slide
• na drugom slajdu = on the second slide

Since slajd is masculine singular, the forms are:

• prvom
• drugom

Je is the 3rd person singular present form of biti (to be). Here it helps form a passive construction:

• projekt je objašnjen = the project is explained / has been explained

So the structure is:

• projekt = subject
• je = is
• objašnjen = explained

Because projekt is singular, the verb is also singular: je.

Objašnjen is a past passive participle, not an ordinary simple past form.

It comes from the verb objasniti = to explain.

So:

• objasniti = to explain
• objašnjen = explained

In projekt je sažeto objašnjen, the meaning is passive:

• The project is/has been explained concisely

The participle agrees with projekt, which is masculine singular, so we get:

• masculine singular: objašnjen
• feminine singular: objašnjena
• neuter singular: objašnjeno
• masculine plural: objašnjeni, etc.

Sažeto means concisely, briefly, or succinctly.

Here it is an adverb, describing how the project is explained:

• projekt je sažeto objašnjen = the project is explained concisely

It is related to the adjective sažet = concise, condensed.

Compare:

• sažet opis = a concise description
• sažeto objašnjen = explained concisely

So even though it looks similar to an adjective form, in this sentence it functions adverbially.

Because the subject in the second clause is primjeri = examples, which is plural.

Compare:

• projekt je... = the project is... (singular)
• primjeri su... = the examples are... (plural)

So:

• je = is
• su = are

Croatian verbs agree with the subject in number.

This is mainly because su is a clitic. Croatian clitics usually appear in the second position in the clause.

So in:

• a na drugom su primjeri

the phrase na drugom comes first, and the clitic su comes immediately after that first unit.

This word order is very natural in Croatian.

You could think of the structure as:

• a | na drugom | su | primjeri

Croatian word order is more flexible than English, but clitic placement is one of the big things that controls it.

Yes, in a way. The noun slajdu is omitted because it is obvious from the context.

Full version:

• a na drugom slajdu su primjeri

Shorter natural version:

• a na drugom su primjeri

English does this too sometimes:

• On the first slide the project is explained, and on the second are examples.

That sounds slightly formal in English, but the idea is similar: the noun is left out because it is already known.

Primjeri is nominative plural.

Singular:

• primjer = example

Plural:

• primjeri = examples

Here it is the subject of the clause:

• na drugom su primjeri = on the second slide are examples

So it must be in the nominative.

Not exactly. A often means and, while, or whereas, and sometimes it can feel a bit like but if there is contrast.

In this sentence:

• Na prvom slajdu projekt je sažeto objašnjen, a na drugom su primjeri.

A links two parallel pieces of information:

• on the first slide: the project is explained
• on the second slide: there are examples

So here a is best understood as something like:

• and
• while
• whereas

It is not a strong contradiction; it is more of a contrast between two different slides.

Yes. A fuller, more explicit version would be:

• Na prvom slajdu projekt je sažeto objašnjen, a na drugom slajdu su primjeri.

You could also say:

• Na prvom slajdu sažeto je objašnjen projekt, a na drugom su primjeri.

That version is also grammatical, but the original sounds very natural and straightforward.

So the original sentence is already idiomatic; Croatian simply allows some flexibility in word order and often omits repeated words when the meaning is clear.