Why do cats so enjoy digging their claws into my legs and how can they be trained to sit on my lap without doing this?

Sometimes they do this when they feel unsafe or as if they are going to slip off your knee.

Maybe if you made sure that your knees were together and that the cat was in the middle of your lap it would help.

Also a throwback to kittenhood when he/she was nursing and would knead mum's tummy to keep the milk flowing. A sign of contentment. We just lift their paws up and take their attention from what they were doing. Who can get inside the mind of a cat???

Sign of affection, m'dear. My cat does this whilst purring obscenely loud and offering a sinister glare.

Boggles the mind, but I just keep quiet and don't move a muscle.

I think I vaguely remember a BBC documentary called something like "The Tiger in Your Living Room".
The answer given there was similar to Jane's in that they said domestic cats had arrested development due to our cosseting of them so exhibit kittenish behaviour into their adulthood.
I'm still not sure I fully understand how that continual purring sound is generated though.

In 8 out of 10 cases a cat's pole will do the trick.

declawing them

Declawing them is cruel, it's like cutting out our fingernails.

I'm not sure if they actually enjoy doing that, but they do it to get comfortable, like some people fluff up pillows. Or if they are falling off you. If you don't like their claws digging into you, either cut them (I do that all the time to my cat) or don't let them sit on your lap. You can try lifting their paws up until they learn you don't like the sharp stabbing pains in your legs.