Krom tio, ke la publiko aplaŭdis, ĝi ankaŭ laŭdis kostumojn.

It means apart from the fact that or besides the fact that.

• krom = apart from, besides
• tio = that
• ke = that, introducing a clause

So Krom tio, ke la publiko aplaŭdis... sets up one fact, and then the main clause adds another fact.

Ke introduces a full clause: la publiko aplaŭdis.

Without ke, tio would just stand alone as that. With ke, Esperanto can specify what that is:

• tio = that
• ke la publiko aplaŭdis = that the audience applauded

So tio, ke... is like the fact that...

The commas separate the introductory clause from the main clause.

The structure is:

• Krom tio, ke la publiko aplaŭdis = introductory subordinate idea
• ĝi ankaŭ laŭdis kostumojn = main clause

Esperanto often uses commas to mark clause boundaries clearly, sometimes more regularly than English does.

Because the second clause needs its own subject.

In Esperanto, you normally do not leave out the subject of a finite verb just because it was mentioned in an earlier clause. So after ...la publiko aplaŭdis, the next clause still needs a subject for laŭdis, and that subject is ĝi.

So:

• la publiko aplaŭdis
• ĝi ankaŭ laŭdis kostumojn

Because la publiko is grammatically singular.

Even though the public or the audience refers to many people in real life, the noun publiko is a singular collective noun. In Esperanto, pronouns usually follow grammatical number, so a singular noun takes a singular pronoun:

• la publiko → ĝi

If you wanted a clearly plural noun, you could say something like la spektantoj, and then you would use ili.

They are related, but not the same.

• aplaŭdi = to applaud, to clap
• laŭdi = to praise

So aplaŭdi is specifically the act of clapping, while laŭdi is a broader idea of giving praise or approval. In this sentence, the audience did both: they applauded, and they also praised the costumes.

Because it is a plural direct object.

Breakdown:

• kostumo = costume
• kostumoj = costumes
• kostumojn = costumes as the direct object

In Esperanto:

• -j marks plural
• -n marks the accusative, usually the direct object

Since laŭdi takes a direct object, kostumojn needs -n.

Because the sentence does not have to specify the costumes.

Without la, kostumojn means costumes in a general or indefinite sense. If the speaker means specific, already known costumes, la kostumojn would also be possible.

So the difference is roughly:

• kostumojn = costumes
• la kostumojn = the costumes

Which one is best depends on context.

Ankaŭ means also or too.

Here it shows that the second action is an addition to the first one:

• the audience applauded
• it also praised the costumes

Its placement before laŭdis is normal and natural.

Yes, in some contexts krom can mean either besides/apart from or except. Context decides which meaning is intended.

In this sentence, the presence of ankaŭ strongly points to the additive meaning:

• besides that the audience applauded, it also praised the costumes

So here it is clearly not an exception.

Yes. A very common alternative would be:

La publiko ne nur aplaŭdis, sed ankaŭ laŭdis kostumojn.

That means The audience not only applauded, but also praised the costumes.

Your original sentence is grammatical, but this alternative may feel more direct to many learners because the not only ... but also structure is easy to recognize.