The short answer
There is currently no legal requirement in England to keep records of home education.
That is the whole legal position, and you can stop reading here if that is all you wanted.
Why keep one anyway
Three reasons, none of them about anybody else.
You forget. By March you will not remember what you did in October. A child's progress is slow to see day to day and obvious over a term, but only if you can look back.
It answers the 9pm question. "Are we doing enough?" is almost impossible to answer from memory and almost easy to answer from a page of dated entries.
It shows the child. A record of work over six months is the best evidence you will ever have of what your child can do, in your hands, for you to show or not show as you choose.
What goes in it
Keep it small enough that you will actually do it.
- A line a day, or a few lines a week. What you covered. Not a lesson plan, a note.
- Samples of work. A photo of a page. One a week is plenty. Date it.
- Reading. Titles, roughly when. Reading is the single most under-recorded thing families do.
- The things that are not lessons. The museum trip, the cooking, the argument about whether the moon is a planet. Education is not only what happens at the table.
- What was hard. This is the entry most people skip and the one that is most useful later, because it is where the next plan comes from.
What to leave out
Do not write it for an imagined reader. Do not grade it. Do not turn it into a timetable. The record is a mirror, not a performance.
Whose record it is
Yours. Keep it somewhere you control, in a format you can export, and delete anything you like whenever you like. Nobody has a right to it. If you ever choose to show part of it to someone, that is your decision and yours alone.
A note on the register
You may have read that a register of children not in school is coming. It is, but the relevant provisions are not yet in force, and keeping records is not something the new Act requires of you. The full position is set out in what is actually changing.