"I am ten years old."
さい (歳 or 才) is the counter for age, so it means "years old". So, "I'm 10 years old" is 十さいです and "I'm 14 years old" is 十四さいです
Of course, in English, it's very natural to leave out the "years old" and just say "I'm 10". However, this doesn't transfer over to Japanese. You must have さい if you are talking about someone's age.
Yes, you might be understood in the right context, but it wouldn't make sense for Duo to allow/teach it because it's not natural or correct to leave it out, unlike many other aspects of Japanese grammar.
As for why this is, this is just a guess, but my theory is that 歳 falls under the category of "counter" and Japanese is very particular about using counters.
The audio sounds to me like, "jussai desu".
Wiktionary does not have an entry for 十歳, but it has an entry for 二十歳, and it gives the pronunciation as "nijussai", also suggesting "jussai" for 十歳.
The Wiktionary entry has a usage note that says, "Jōyō kanji recognized only じゅう (jū) and じっ (ji') as readings of 十 until 2010, even though the latter is actually pronounced じゅっ (ju') today and NHK accepted it already in 1966." This is slightly unclear to me, but is the っ here being used to represent the gemination of the next consonant, and if so, it saying that "jussai" is the current pronunciation, albeit a relatively recent innovation?
Google Translate gives the transliteration, "Jū-saidesu," for the sentence, even though their audio sounds much like Duolingo's. Is their transliteration an alternative, valid pronunciation, or are they stuck in the past? Or just wrong?
Other Duolingo users commenting on this page have given "Ju sai", "jyu sai", and "Juusai", and none of them have been corrected.
TL:DR: Most sources seem to suggest "jussai" as the pronunciation for 十歳, but there is disagreement from at least one external source and from several comments here. But is "jussai" definitely correct?
"sai" is on-yomi used in compound words to mean 'year' and for the counter for years, "toshi" is the kun-yomi and the noun "year", though for the noun the kanji 年 is typically used instead. The TTS tends to default to kun-yomi readings when a kanji is isolated because those are the readings a kanji usually takes in that context.