て形
te-kei
The て form means nothing. It is a hook.
The picture て turns the end of a verb into a small hook. Whatever comes next hangs off that hook. て leaves the sentence open on a hook; た brings the stamp down and closes it.
Where the picture shows up in て形
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the 'and then' hook
手を洗って、ご飯を食べます。
te o aratte, gohan o tabemasu
I wash my hands and then eat.
The hand-washing panel ends in a hook, and the eating panel hangs from it — two frames on one string.
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the 'by doing' hook
電車に乗って会社まで行きます。
densha ni notte kaisha made ikimasu
I take the train to work.
You grab the hook hanging from the train ceiling and get carried all the way to the office.
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the 'because' hook
遅れてすみません。
okurete sumimasen
Sorry I'm late.
A card reading 'late' hangs on the hook, with 'I'm sorry' swinging from it underneath.
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the hook for a request
ここに名前を書いてください。
koko ni namae o kaite kudasai
Please write your name here.
書いて leaves a hook, and ください is the hand that catches it and pulls the pen down to the paper.
ください always follows the て form. There is no 書きください.
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the hook for permission
先に帰ってもいいですか。
saki ni kaette mo ii desu ka
Is it all right if I leave first?
帰って leaves a hook, an 'OK' tag is hung on it, and the door swings open.
All of it in one picture
Add て and the sentence refuses to stop: it puts out a hook instead. Hang the next action on it and you get sequence; hang the vehicle on it and you get means; hang what happened first and you get a reason. Hang ください and it is a request; hang もいい and it is permission. So there is nothing to memorise five times over — one hook, and only the thing you hang there changes.