what a person mentions while preaching is,or rather,should be explanation of Quran,such talk can be continued
It is highly recommended to answer the call for prayer (adhan) by saying what the mua'adhdhin says based on:
The Messenger of Allah (ﷺ) said: When the Mu'adhdhin says: Allah is the Greatest, Allah is the Greatest, and one of you should make this response: Allah is the Greatest, Allah is the Greatest; (and when the Mu'adhdhin) says: I testify that there is no god but Allah, one should respond: I testify that there is no god but Allah, and when he says: I testify that Muhammad is the Messenger of Allah, one should make a response: I testify that Muhammad is Allah's Messenger. When he (the Mu'adhdhin) says: Come to prayer, one should make a response: There is no might and no power except with Allah. When he (the Mu'adhdhin) says: Come to salvation, one should respond: There is no might and no power except with Allah, and when he (the Mu'adhdhin) says: Allah is the Greatest, Allah is the Greatest, then make a response: Allah is the Greatest, Allah is the Greatest. When he (the Mu'adhdhin) says: There is no god but Allah, and he who makes a re- sponse from the heart: There is no god but Allah, he will enter Paradise. (See for example in Sahih Muslim)
When you hear the Mu'adhdhin, repeat what he says, then invoke a blessing on me, for everyone who invokes a blessing on me will receive ten blessings from Allah; then beg from Allah al-Wasila for me, which is a rank in Paradise fitting for only one of Allah's servants, and I hope that I may be that one. If anyone who asks that I be given the Wasila, he will be assured of my intercession. (See for example in Sahih Muslim)
Some scholars took the hadith literally and considered repeating the words of the mu'adhdhin as rather compulsory these are in first place the Dhahiri's and the Hanafi's.
While the majority consideres repeating the words of the mu'adhdhin as highly recommended (mustahhab), as the prophet () himself heard a person calling for prayer and said: "He is following the fitrah" and giving comments and there's no mentioning that he repeated the words of the mu'adhdhin:
The Messenger of Allah (ﷺ) used to attack the enemy when it was dawn. He would listen to the Adhan; so if he heard an Adhan, he stopped, otherwise made an attack. Once on hearing a man say: Allah is the Greatest, Allah is the Greatest, the Messenger of Allah (ﷺ) remarked: He is following al-Fitra (al-Islam). Then hearing him say: I testify that there is no god but Allah. there is no god but Allah, the Messenger of Allah (ﷺ) said: You have come out of the Fire (of Hell). They looked at him and found that he was a goat herd. (See for example in Sahih Muslim)
You may also find a secondary evidence in Muwatta' Malik saying that people used to speak while the mua'adhdhin was calling for prayer at the time of 'Omar ibn al-Khattab:
Yahya related to me from Malik from Ibn Shihab that Thalaba ibn Abi Malik al Quradhi informed him that in the time of Umar ibn al Khattab they used to pray on the day of jumua until Umar came out, and when Umar came out and sat on the mimbar and the muadhdhins called the adhan, they would sit and talk, and then when the muadhdhins were silent and Umar stood to give the khutba, they would pay attention and no-one would speak .
Ibn Shihab said, "The imam coming out stops prayer and his speaking stops conversation." (Al-Muwatta')
Imam an-Nawawi therefore said in his al-Majmo' المجموع شرح المهذب:
In the following I'm translating from Arabic, as these translations are of my own take them carefully!
مذهبنا أن المتابعة سنة ليست بواجبة وبه قال جمهور العلماء ، وحكى الطحاوي خلافا لبعض السلف في إيجابها وحكاه القاضي عياض . (Source)
Our madhhab (view) is that repeating the words of the adhan is sunnah and not wajib and this is the view of the majority of scholars, and at-Tahawi quoted a difference of opinion from some of the salaf in answering and al-Qadi 'Iyad quoted it too.
Therefore the matter is wide open one may stop his lesson to repeat the words of adhan and get the rewards or continue the lesson. As for the audience repeating the words of adhan is highly recommended unless one is afraid to miss some important information.
My teacher (May Allah have mercy on him) used to say learning (studying) is fard (kifaya) while a sunnah prayer or adhan is sunnah ... this means one should set priorities.
For example if you heard the call for prayer while you were reading or reciting Qur'an there are two sunnah (recommended) acts you could get rewards from, but as you may continue your recitation after an interruption, while you can only answer the prayer call at this moment: interrupting your quran recitation or reading should be your choice as you will else miss the rewards of repeating the adhan words.
Some related fatwas and further references: