Although I can absolutely see why this sentence is confusing, I also find it to be rather natural in my dialect of American English. When you see "to do" followed by an infinitive, it might help to mentally insert "in order" between the infinitives. Examples: 1) Working hard is what I have to do (in order) to pay my bills. 2) Walking dogs is what I like to do (in order) to stay healthy. In none of these cases do we say "in order". While doing so might make the sentence more easily understood to non native speakers, it would also make it sound stuffy and awkward to us. Additionally, this construction should only be used in spoken English. For formal or academic writing, these sentences would need to be reworded. 1) I have to work hard to pay my bills. 2) I like to walk dogs to stay healthy. 3) It was really exciting to see HOW the people at the museum were able to clean the silver pieces. They were then able to find out exactly what each piece is. Or It was really exciting to see the methods the people at the museum use to clean the silver pieces and discover what they are. I hope this helps.

You've got a point. Often, learning and testing materials prepared by non-native speakers suffer from poor wording like this. Using two infinitives back-to-back isn't always a bad thing, but here it's part of a very poorly-worded sentence. Rather than "...what (they) were able to do to clean the pieces of silver and find out exactly what they are", one way to say it more clearly could be "...how well (they) had cleaned the pieces of silver to make them easier to identify."

Infinitives can act in three different ways: as nouns, as adjectives, and as adverbs. Since they are so enormously flexible, it takes some practice to recognize how they are being used. Your sentence contains FOUR infinitive phrases. Making things even more complicated, these phrases are nestled one inside another so that understanding the sentence requires that you unpeel it like an onion. Let's give names to these phrases to make the discussion easier: A = "to clean the pieces of silver" (adverbial phrase that modifies the verb "do") B = "(to) find out exactly what they are" (adverbial phrase that modifies the verb "do") C = "to do A and B" (adverbial phrase that modifies the adjective "able") D = "to see what the people at the museum were able C" (noun phrase) D merely tells you the meaning of "it". In fact, you can eliminate the word "it" and rewrite the whole sentence by moving the noun phrase to the beginning: "D was really exciting." To see that A and B act as adverbs that modify "do", substitute more obvious adverbs like "enthusiastically" and "cheerfully": "...what they were able to do enthusiastically and cheerfully" Finally, let's look at "to do". It is an adverb that modifies the adjective "able". The adjective "able" describes the noun "people". They were "able people". The adverb "to do" tells you in what way, or how, they were able. Let's look at a simpler example: "The birds were able" ("able" is an adjective describing "birds") "The birds were able to fly" (the adverbial infinitive "to fly" tells HOW they were "able") "the birds were able to fly home" ("home" is an adverb that further modifies "to fly" by answering a WHERE question)

It’s a perfectly natural and correct thing to say. Look up ‘to have to do something’. Your suggestion doesn’t have the same meaning. They have to do something to clean the pieces. This something is what they had to do to clean the pieces. I had to lose weight to fit into my old clothes. … the weight I had to lose to fit… She had to earn money to pay the rent. … the money she had to earn to pay the rent.