[This is] a Book revealed to you, [O Muḥammad] - so let there not be in your breast distress therefrom - that you may warn thereby and as a reminder to the believers.
In the Name of Allah, the Most Gracious, the Most Merciful.
كِتَـبٌ أُنزِلَ إِلَيْكَ
((This is the) Book (the Qur'an) sent down unto you (O Muhammad)), from your Lord,
فَلاَ يَكُن فِى صَدْرِكَ حَرَجٌ مِّنْهُ
(so let not your breast be narrow therefrom,) meaning, having doubt about it according to Mujahid, Qatadah and As-Suddi. It was also said that the meaning here is: `do not hesitate to convey the Qur'an and warn with it,'
In the statement فَلَا يَكُن فِي صَدْرِكَ حَرَجٌ appearing in the first verse, the address is to the Holy Prophet . ؓ and he has been told: This Qur'an is the Book of Allah sent down to you. This should not cause any constraint on your heart. The word: حَرَج (haraj) translated here as ` constraint' means that ` you should have no anguish or apprehension in conveying the Qur'an and its injunctions lest people belie it and hurt you.' (As reported from Abu Al-` Aliyah - Mazhari)
The hint given here is that Allah who has sent down this Book on you has also made arrangements that you shall remain protected and that takes care of any anguish on your part. Some commentators have said that ` haraj' or ` constraint on the heart' refers to the constraint experienced by the Holy Prophet ؓ who, because of his affectionate concern for people, felt pain when they would not believe despite having heard the Qur'an and its injunctions. To offset this constraint, the Holy Prophet ﷺ has been told that the duty with which he has been obligated is simply to make the call and convey the message. Once this is done, it is not his responsibility to see who becomes a Muslim and who does not. Therefore, there was no reason for him to be anxious.
Tazkirul Quran
Maulana Wahiduddin Khan
The Book of God is, in essence, a book of guidance. But as such it is of value only to the few who have kept their consciences alive. For the others it amounts only to a warning of the impending grievous fate towards which they are moving because of their arrogance. The preacher of truth is concerned to see that what he sees as the absolute truth is being rejected by the majority of the people who consider it a falsehood. What is paramount in his eyes is treated by people with unconcern, as if it had no substance, as if it were completely unreal.
Uthmani text of the Madinah mushaf (King Fahd Complex). Translation: Saheeh International. Tafsir via Quran.com.