Doctor here:
When you lean on your arm you compress the nerves. This irritates them. The pins and needles sensation is not primarily from a lack of blood supply, despite what other have written here, although it can be a contributing factor.
When the compression stops, the nerves begin to work again. They are irritated so they report this to your brain in the form of pain. After a while the irritation ends and things go back to normal.
When a numbing agent is used, it simply blocks the transmission of pain rather than compressing and irritating the nerves.
There’s more to it than this, but this is ELI5.
EDIT: some typos.
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