God reveals in the Qur'an that Muslims must believe in our Prophet Muhammed (may God bless him and grant him peace) and in all the prophets sent before him and in everything sent down to those prophets. In verse 4 of Surat al-Baqara, our Lord says that Muslims are "those who believe in what has been sent down to you and what was sent down before you..."
As revealed in the Qur'an, the Gospel, Torah, Psalms and the pages of Abraham are books sent down to prophets who lived in the past. Some of these books have disappeared, while others have been altered or distorted. But they nevertheless contain a great many correct pronouncements belonging to the true faith. Muslims analyze these according to the Qur'an and the Sunna [teachings of the Prophet Muhammad (may God bless him and grant him peace)] and decide that they are correct statements compatible with verses and the hadith. Therefore, statements in the Torah and the Gospel that conform to the Qur'an and the Sunna are statements that Muslims can benefit from. But the essential precondition is for these statements to be in agreement with the verses of the Qur'an and the hadiths.
God revealed that these books were guides to the people living at the time they were sent. Allah tells in the verses:
He has sent down the Book to you with truth, confirming what was there before it. And He sent down the Torah and the Gospel, previously, as guidance for mankind, and He has sent down the Furqan. (Surah Al 'Imran, 3-4)
In other verses about the Gospel our Lord reveals:
And We sent Jesus son of Mary following in their footsteps, confirming the Torah that came before him. We gave him the Gospel containing guidance and light, confirming the Torah that came before it, and as guidance and admonition for those who have taqwa [righteousness]. The people of the Gospel should judge by what Allah sent down in it. Those who do not judge by what Allah has sent down, such people are deviators. (Surat al-Ma'ida, 46-47)
Then We sent Our Messengers following in their footsteps and sent Jesus son of Mary after them, giving him the Gospel. We put compassion and mercy in the hearts of those who followed him.They invented monasticism–We did not prescribe it for them–purely out of desire to gain the pleasure of Allah, but even so they did not observe it as it should have been observed. To those of them who had faith We gave their reward but many of them are deviators. (Surat al-Hadid, 27)
This book you are reading only uses some of the extracts from the Gospel that are compatible with the Qur'an and the Sunna of our Prophet Muhammad (may God bless him and grant him peace).