What is the significance of Dhṛtarāṣṭra’s saying māmakāḥ in 1.1?

It illustrates his way of thinking. He separates the Pandavas from the Kuru heritage. Thinking that only his sons are the rightfull heirs to the Kuru throne and not Pandu's sons.

Why is Dhṛtarāṣṭra asking to Sañjaya: what did my sons and the sons of Pāṇḍu do, being desirous to fight?

Because he was highly doubtful about the possibility of his sons' ultimate victory. In his doubt, he inquired from his secretary Sañjaya, "What did my sons and the sons of Pāṇḍu do?" He was confident that both his sons and the sons of his younger brother Pāṇḍu were assembled in that Field of Kurukṣetra for a determined engagement of the war. Still, his inquiry is significant. He did not want a compromise between the cousins and brothers, and he wanted to be sure of the fate of his sons on the battlefield. Because the battle was arranged to be fought at Kurukṣetra, which is mentioned elsewhere in the Vedas as a place of worship—even for the denizens of heaven—Dhṛtarāṣṭra became very fearful about the influence of the holy place on the outcome of the battle. He knew very well that this would influence Arjuna and the sons of Pāṇḍu favorably, because by nature they were all virtuous. Sañjaya was a student of Vyāsa, and therefore, by the mercy of Vyāsa, Sañjaya was able to envision the Battlefield of Kurukṣetra even while he was in the room of Dhṛtarāṣṭra. And so, Dhṛtarāṣṭra asked him about the situation on the battlefield.

Both the Pāṇḍavas and the sons of Dhṛtarāṣṭra belong to the same family, but Dhṛtarāṣṭra's mind is disclosed herein. He deliberately claimed only his sons as Kurus, and he separated the sons of Pāṇḍu from the family heritage. One can thus understand the specific position of Dhṛtarāṣṭra in his relationship with his nephews, the sons of Pāṇḍu.

Dhṛtarāṣṭra was the eldest son of the Kuru dynasty, but he was blind, bodily defect. Therefore he was not awarded the throne. His next brother, Pāṇḍu, he was offered the throne, but he died at a very early age, a young man. When these sons of Pāṇḍu, the five sons, Yudhiṣṭhira Mahārāja, at that time not Mahārāja, Yudhiṣṭhira, Bhīma, Arjuna, Nakula, Sahadeva, they were very small children, so they were taken care of by Dhṛtarāṣṭra and other elderly familymembers like Bhīṣmadeva. He was the grandfather of the Pāṇḍavas. Bhīṣma was elder brother of Dhṛtarāṣṭra's father. He was so old. 

Actually, the kingdom belonged to Bhīṣma, but he remained a brahmacārī, he did not marry. There was no issue of Bhīṣmadeva. Therefore his nephews, Dhṛtarāṣṭra and Pāṇḍu, they were inheritor.

Now, after the death of Pāṇḍu, there was conspiracy. Dhṛtarāṣṭra thought "Actually, this is my kingdom. Now, somehow or other, I could not get it. Now my brother is dead. So if I do not inherit, why not my sons?" This is called politics. Enviousness and jealousy is always found in politics. This is the nature of this material world. You cannot avoid it. Spiritual world means just the opposite. There is no politics. There is no jealousy. There is no enviousness. That is spiritual world. And material world means politics, jealousy, diplomacy, enviousness, so many things. This is material world. So even in the heavenly planets, these things are there, politics. Even in animal kingdom, these politics are there. This is the nature. Matsaratā. Matsaratā means enviousness. One man is envious of another man. It doesn't matter, even they are brothers or family members like Dhṛtarāṣṭra and Pāṇḍu.