We don’t know precisely what Nephi meant by "the language of the Egyptians," but quite a bit has been written on this topic. We have lots of archaeological evidence of the presence of Egyptian culture and influence in Israel.
Jerusalem has always been a borderland between the two big river valleys of Egypt and Mesopotamia. Nebuchadnezzar would try to push his power farther south. Other times, the Egyptians would drive things farther north. People living in Jerusalem always had to keep a lookout in both directions. For strategic reasons and for other purposes, we know that Israelites had contact, trade and alliances with these super powers on both sides.
A case in point here is that the Lord called multiple prophets to stand in the streets and prophesy that Jerusalem would be destroyed. Urijah was one of these prophets. When he was chased out of the city, where did he run? To Egypt. This is evidence of contact between people in Egypt and people in Israel. Jewish people must have had a community in Egypt where Urijah thought he could flee and find refuge. But what happened to Urijah? As further evidence of an even higher level of relationship between Egypt and Israel, rulers in Jerusalem sent a delegation to Egypt, extradited Urijah, brought him back to stand sentencing and ignominious execution (see Jeremiah 26).
More evidence of Egyptian influence in Israelite culture is a papyrus that was found in Egypt called the Amherst Papyrus, which is actually a Hebrew Psalm that is written in demotic Egyptian characters. So, in this case, one has the learning of the Hebrews (the literature of the Hebrews) written in Egyptian characters, and that is an eighth-century B.C. text. So, some of these people were bilingual, and they could use different scripts to write down the sounds and the words of other languages. Today, that would be a bit like taking a Japanese word and spelling it with Roman letters. If you don’t know both languages, you can’t make much sense of it.
What was the nature of record-keeping in Lehi’s world? Did the Egyptians write? Absolutely, all the time! Every mummy, every tomb, every temple wall, every palace had writing all over it. Now, we do not find that kind or volume of writing in the archaeological remains of the Israelites, but it is not hard to believe that Israelites knew plenty about writing. They encountered it. People traveled through Jerusalem. They had merchants coming and going. Lehi may well have traveled to Egypt himself, where he learned how to read and write Egyptian. Lehi was a reasonably sophisticated person. As a caravan owning merchant, he must have been literate and able to keep records and communicate with people in different lands.
Of course, writing was not just used for business records. Most of what was written in Egyptian were sacred religious texts, like the Book of the Dead. If you were wealthy, you had the whole Book of the Dead buried with you or parts of it copied onto the walls of your tomb. But even the poor had some portion of the Book of the Dead written out for them. Why? The Book of the Dead gave the road map of the next life. To pass the tests one would encounter in the next life, you needed that information. It contained the passwords, the keys and the language they needed to use in order to express eternal loyalty and much more.
In Israel, what did they mostly write? Archaeologists have found a few little messages scrawled on potsherds saying, "Help! Our city is being over-run, send soldiers." Recently, they discovered one of the earliest Hebrew writings, and it turns out it appears to be a shopping list. Go to the market and get this, this, and this. That tells us something about who could write: an ordinary person writing ordinary words. And that tells us that writing was more common than many have previously thought.
Even so, the main things the Israelites captured in writing were the teachings of the prophets and the words of the law. They needed the law, especially in the temple, so that the priests would know the order of sacrifice, the rules of purity and other things that they had to do. One of the jobs of the Levites was to maintain the records—to copy them. The records had to be copied over and over again because they wore out. Papyrus wears out about every 30 years in regular use. So, scribes were needed and were trained to do this writing.
Other recently found Hebrew texts that have survived are two small silver scrolls. They are each about an inch long and have archaic Hebrew letters, written on very thin silver and rolled up. One was worn as a necklace by a young girl who was buried around 610 BC—the time of Lehi. The text on the scroll includes part of the priestly blessing found in Numbers 6:24–26, "The Lord bless and keep thee. The Lord make his face shine upon thee … and give thee peace." Coming from Numbers and dating to Lehi’s day, this was a real shock to a lot of biblical scholars when it was found. Here was evidence of people writing religious texts, writing on metal, and drawing upon scriptures at that early date, well before the time when the Jews were taken to Babylon.
This was a part of Lehi’s world, and Lehi was a scribe who could read the plates of brass and wrote his own book of Lehi. Additionally, Nephi would have spent his days as the youngest son often did, not out farming and not out learning military tactics, but in learning language, writing records and reading scriptures.
Book of Mormon Central, "Did Ancient Israelites Write in Egyptian? (1 Nephi 1:2)," KnoWhy 4 (January 5, 2016).
Pike, Dana M. "Israelite Inscriptions from the Time of Jeremiah and Lehi." In Glimpses of Lehi’s Jerusalem, edited by John W. Welch, David Rolph Seely and Jo Ann H. Seely, 193–244. Provo, UT: Foundation for Ancient Research and Mormon Studies, 2004.
Welch, John W. "The Trial of Jeremiah: A Legal Legacy from Lehi’s Jerusalem." In Glimpses of Lehi’s Jerusalem, edited by John W. Welch, David Rolph Seely and Jo Ann H. Seely, 337–356. Provo, UT: Foundation for Ancient Research and Mormon Studies, 2004.