Why does the first-person pronoun “we” appear in select instances in the latter half of the narrative of the Acts of the Apostles. Here is the explanation of Irenaeus, the late second-century bishop of Lyon, in his famous treatise On the Detection and Overthrow of the So-called Knowledge or Against Heresies 3.14.1:
“But that this Luke was inseparable from Paul, and his fellow-labourer in the Gospel, he himself clearly evinces, not as a matter of boasting, but as bound to do so by the truth itself. For he says that when Barnabas, and John who was called Mark, had parted company from Paul, and sailed to Cyprus,we came to Troasand when Paul had beheld in a dream a man of Macedonia, saying,Come into Macedonia, Paul, and help us,immediately,he says,we endeavoured to go into Macedonia, understanding that the Lord had called us to preach the Gospel unto them. Therefore, sailing from Troas, we directed our ship’s course towards Samothracia.And then he carefully indicates all the rest of their journey as far as Philippi, and how they delivered their first address:for, sitting down,he says,we spoke unto the women who had assembled andcertain believed, even a great many. And again does he say,But we sailed from Philippi after the days of unleavened bread, and came to Troas, where we abode seven days.And all the remaining [details] of his course with Paul he recounts, indicating with all diligence both places, and cities, and number of days, until they went up to Jerusalem; and what befell Paul there, how he was sent to Rome in bonds; the name of the centurion who took him in charge and the signs of the ships, and how they made shipwreck and the island upon which they escaped, and how they received kindness there, Paul healing the chief man of that island; and how they sailed from thence to Puteoli, and from that arrived at Rome and for what period they sojourned at Rome. As Luke was present at all these occurrences, he carefully noted them down in writing, so that he cannot be convicted of falsehood or boastfulness, because all these [particulars] proved both that he was senior to all those who now teach otherwise, and that he was not ignorant of the truth. That he was not merely a follower, but also a fellow-labourer of the apostles, but especially of Paul, Paul has himself declared also in the Epistles, saying:Demas has forsaken me, and is departed unto Thessalonica; Crescens to Galatia, Titus to Dalmatia. Only Luke is with me. Fromthis he shows that he was always attached to and inseparable from him. And again he says, in the Epistle to the Colossians:Luke, the beloved physician, greets you.But surely if Luke, who always preached in company with Paul, and is called by himthe beloved,and with him performed the work of an evangelist, and was entrusted to hand down to us a Gospel, learned nothing different from him (Paul), as has been pointed out from his words, how can these men, who were never attached to Paul, boast that they have learned hidden and unspeakable mysteries?” 
 Translated by Alexander Roberts and William Rambaut. From Ante-Nicene Fathers, Vol. 1. Edited by Alexander Roberts, James Donaldson, and A. Cleveland Coxe. (Buffalo, NY: Christian Literature Publishing Co., 1885.) Revised and edited for New Advent by Kevin Knight. <http://www.newadvent.org/fathers/0103.htm>. I have removed the insertions of the biblical references.