Can you please explain why some people would argue that the Gospel of Luke and Acts were written by the same person?
The fact that both Books were written by the same individual known as Luke is extracted from the first few verses of both Books.
Luke 1:1 through4 KJV Forasmuch as many have taken in hand to set forth in order a declaration of those things which are most surely believed among us, Even as they delivered them unto us, which from the beginning were eyewitnesses, and ministers of the word; It seemed good to me also, having had perfect understanding of all things from the very first, to write unto thee in order, most excellent Theophilus, That thou mightest know the certainty of those things, wherein thou hast been instructed.
Acts 1:1 KJV , The former treatise have I made O Theophilus, of all that Jesus began both to do and teach,
Both books were written to one Theophilus, and The former treatise have I made O Theophilus indicates that Acts is a follow-up to the Book of Luke.
These two commentary excerpts deal with these introductions to the books.
Forasmuch as many have taken in hand,.... From hence, to the end of Luk_1:4 is a preface of the evangelist to his Gospel, setting forth the reasons of his writing it; and which he wrote and sent to the excellent Theophilus, for the further confirmation of him in the faith of Christ. It seems that many had took in hand, or attempteo set forth in order a declaration of those things which are most surely believed among us; that is, they undertook to write and publish a very particular and exact narrative of the birth, life, actions, doctrines, miracles, sufferings, death, resurrection, and ascension of Jesus Christ; things which Luke, and other Christians, had the fullest and strongest evidence, and were confidently assured of, and most firmly believed, even with a full assurance of faith. By these many, he cannot mean the authentic historians of evangelical facts, as Matthew and Mark; for they two cannot, with any propriety, be called many; and besides, it is not so very clear and certain a point, that they had, as yet, wrote their Gospels; nor would this evangelist suggest any deficiency, weakness, and inaccuracy in them, as he seems to do: nor does he intend such spurious writers as the authors of the Gospels according to the Nazarenes, Hebrews, and Egyptians; of Nicodemus, Thomas, Matthias, and of the twelve apostles; and still less, the Gospels of Cerinthus, Basilides, and other heretics; since these would not have passed without a censure from him, for the falsehood, fabulous, and trifling stuff in them, as well as for the wicked and heretical opinions propagated by them; and besides, these pieces were not extant when this Gospel was written: but he seems to design some honest and well meaning Christians, who undertook to write, and did write an account of the above things, which were firmly believed by all; and which they took from the apostles, and first ministers of the Gospel, from their sermons and discourses, and from conversation with them; and which they committed to writing, partly to help their own memories, and partly for the benefit of others; in which, no doubt, they acted an upright part, though attended with weakness: wherefore, the evangelist does not censure them as false, wicked, and heretical, nor approve of them as divine and perfect for though they honestly meant, and designed well, yet there might be many things collected by them, which were impertinent, and not proper to be transmitted to posterity; and what might be wrote with great inaccuracy and deficiency, and in a style the Holy Ghost thought improper things of this kind should be delivered in: and therefore the evangelist, moved and inspired by the Spirit of God, set about the following work, and under the same influence completed it. The phrase, αναταξασθαι διηγησιν, "to set forth in order a declaration", is as Dr. Lightfoot observes, out of the Talmud (h), agreeably to the Jewish way of speaking, "R. Chasdai said to one of the Rabbins, who was מסדר אגדתא, "setting in order a declaration" before him. &c. or relating in order a story before him. (h) T. Bab. Succa, fol. 53. 1."
Commentary by John Gill
The former treatise have I made,.... Meaning the Gospel written by him the Evangelist Luke, for from that he makes a transition to this, beginning here where he there left off; namely, at the ascension of Christ; see Luk_24:51."
Commentary by Peter Gusik
With further examination of the two books there are similarities in both style and choice of syntax which lend both books to a common authorship.
Hope this helps
There are a few reasons why people argue both of those books are written by Luke.
Both the book of Luke and the book of Acts reference a reader named "Theophilus" in their opening lines. The book of Luke sets up an initial correspondence with Theophilus:
Many have undertaken to draw up an account of the things that have been fulfilled among us, just as they were handed down to us by those who from the first were eyewitnesses and servants of the word. With this in mind, since I myself have carefully investigated everything from the beginning, I too decided to write an orderly account for you, most excellent Theophilus, so that you may know the certainty of the things you have been taught. Luke 1:1-4
The book of Acts begins with a continuation of the correspondence with Theophilus:
In my former book, Theophilus, I wrote about all that Jesus began to do and to teach until the day he was taken up to heaven, after giving instructions through the Holy Spirit to the apostles he had chosen. Acts 1:1-2
Many people believe the "former book" referenced in Acts is the book of Luke.
The "We" Sections of Acts
Three times in the book of Acts, the writer switches to using the pronoun "we" when talking about travelling with Paul. These instances are in Acts 16:10-18, Acts 20:4-21:19, and Acts 27:1-28:30. This use of "we" informs us that the author was indeed a companion of Paul, so the assumption can be made that it is probably someone whom Paul would have referenced in his epistles. This rules out the men who Paul references in his epistles and who also are referenced in the third-person in Acts (Aristarchus, Tychicus, Timothy, and Mark), leaving Luke as a viable candidate. For more information on where Luke would have been based on the "we" passages in Acts, see Luke's Eye-witness Accounts in Acts by Kevin Rogers.
The Muratorian Canon
The Muratorian Fragment is a fragment of a manuscript dating back to around 170 A.D. containing the oldest known list of books of the New Testament. This manuscript confirms which books were in included in the New Testament not incredibly long after they were written. In the fragment, the author says the following in regards to Luke and Acts:
The third book of the Gospel is that according to Luke. Luke, the well-known physician, after the ascension of Christ, when Paul had taken with him as one zealous for the law, composed it in his own name, according to [the general] belief. Yet he himself had not seen the Lord in the flesh; and therefore, as he was able to ascertain events, so indeed he begins to tell the story from the birth of John.
Moreover, the acts of all the apostles were written in one book. For 'most excellent Theophilus' Luke compiled the individual events that took place in his presence — as he plainly shows by omitting the martyrdom of Peter as well as the departure of Paul from the city [of Rome] when he journeyed to Spain. Translation by Hans Lietzmann
So even in more ancient times, Luke was believed to be the author of both Luke and Acts.
You ask, “why some people would argue ...”, implying that other people (many other people? most other people?) would reject that argument. Nevertheless, it is the generally accepted view. This is the first time I have heard of any alternative hypothesis about the authorship of the two books addressed to Theophilus. The only disagreement that you commonly encounter is that between the groups that Ben Witherington calls the “unitarians” and the “separatists” — i.e. between those who regard Luke and Acts as volumes 1 and 2 of a single work, and those who see them as two separate works by the same author.
In addition to what the others said about the introduction to Theophilus, it is also writing style and Grammer. When reading the Greek Text, Luke's is the most difficult/or said I say more advanced than the other books, and Luke and Acts are the same style of writing. Both Luke and Acts. So you got the introduction to Theophilus and the Grammer and writing style.