This also implies what is not mentioned, that this promised boy was born and he was Yahya. There is also the implication that Allah taught him the Book, the Tawrah which they used to study among themselves. The Prophets who were sent to the Jews used to rule according to the Tawrah, as did the scholars and rabbis among them. He was still young in age when Allah gave him this knowledge. This is the reason that Allah mentioned it. Because of how Allah favored him and his parents, He says,
ييَحْيَى خُذِ الْكِتَـبَ بِقُوَّةٍ
(O Yahya! Hold fast to the Scripture the Tawrah.) Means, "Learn the Book with strength." In other words, learn it well, with zeal and studious effort.
حَنَانًا Literal meaning of this word is soft heartedness, compassion and mercy, and these qualities were specially granted to Sayyidna Yahya (علیہ السلام) .
Tazkirul Quran
Maulana Wahiduddin KhanOn ayahs 11–14
Just as the creation of the first human being, without a father or a mother, was a miracle of God, the birth of a child through a father and a mother is also God’s miracle, irrespective of whether the father and mother are old or young. The fact is that it is God who creates man. He created him in the first instance and He creates man even today. The seeming cause is not the real reason, though appearances would suggest otherwise. Zachariah asked for a sign as a token of the mercy God should shower upon him. He was told that, in spite of being healthy, if he could not talk for three continuous days and nights, he should take it to be a confirmation of his wife’s pregnancy. So, when the time came, his tongue would not utter a sound and he become mute. He came out of his place of worship and started exhorting the people by signs to remember God day and night, to worship Him and always obey Him. Probably Zachariah was in the habit of delivering sermons and good advice to people every day. So when he was rendered speechless, he went to the usual meeting place and, as always, gave good advice to the people. But, since he could not speak, he counseled people by signs.
Uthmani text of the Madinah mushaf (King Fahd Complex). Translation: Saheeh International. Tafsir via Quran.com.