How did Egyptians make writing sheets from a swamp reed?

By slicing and hammering. Egyptians took the stem of a wetland sedge, Cyperus papyrus, removed its outer rind and cut the sticky inner pith lengthwise into thin strips about 40 centimetres long. They laid the strips side by side, edges slightly overlapping, put a second layer on top at right angles, and hammered the two layers together while still moist until they mashed into a single sheet. The sheet was dried under pressure and polished with a rounded stone, shell or piece of hardwood; sheets could then be glued together into a long roll.

The oldest papyrus ever found isn't a poem or a letter: it's a work diary. Excavated in 2012 and 2013 at Wadi al-Jarf, an ancient harbour on the Red Sea, the Diary of Merer dates to around 2560–2550 BCE and describes the last years of building the Great Pyramid of Giza. Papyrus was cheap and easy to make, but fragile: it could not be folded without cracking, it suffered from both damp and excessive dryness, and in European conditions a 200-year-old papyrus was considered extraordinary. Its name lives on, though: papyrus is the origin of the word "paper".

Parchment, made from animal skins, gradually overtook it in West Asia and Europe, because it lasted better in moist climates and was easier to write on both sides; by the beginning of the 4th century CE the most important books were made of parchment. The latest certain use of papyrus in Europe is a papal decree of 1057, and in Egypt it was finally replaced by cheaper paper introduced by the Islamic world. Its fragility makes survivors precious: papyri give us the only surviving copy of Menander, the Egyptian Book of the Dead and the mathematical Rhind papyrus.