Learn Dutch: Een euro is honderd cent. - One euro is one hundred cents.

In Dutch, you do not need to add een (a or one) before honderd (hundred) or duizend (thousand). Just saying honderd already means "one hundred".

Why is the word cent singular even though there are a hundred of them?

In Dutch, currencies and units of measurement (like euro, cent, or kilos) usually stay in their singular form when they come directly after a number. Because of this rule, we say honderd cent instead of honderd centen.

Doesn't een mean "a"? Why does the English translation say "One euro"?

The word een can mean both "a" and "one" in Dutch. While you will sometimes see it written with accents as één to strongly emphasize the number "one", the normal een is very commonly used for "one" when the context makes it clear, just like here.

Are the words for currencies like euro and cent ever capitalized?

No. Just like in English, currency names are always written with lowercase letters in Dutch unless they happen to be the very first word of a sentence.