'Отписаться' has meaning "to write letter you must provide with a sole intention to formally fulfill this requirement, but not to solve problem behind it". It's like 'от-вертеться', 'от-прыгнуть' in the sense of 'от-' as 'avoid something with action supplied'. This meaning exists.
But this word is example of single word with opposite meanings (I forgot term for this, english has rich set of these). In this letter 'от-' is used as direction prefix 'from-' and just used in the sense 'there will be letter from me'.