Of the same reason that English does not say: 'My the socks'. When a noun is in the definite form, the definition can either be only the definite article (here plural -orna) or with a pronoun ('mina strumpor'), or some other determining word. But in this sentence: Jag tar av mig strumporna I suspect you mix up 'mig' which has to do with the verb, with 'mina' that definies the noun. We say "ta av sig", reflexive form, to show that it is I who take off my own socks on myself, where as English just say 'take off' and instead put the definition on the noun, as being 'my socks'. I hope this did not mix it up to much for you.
Your English sentence is possible, but then you would have to change the Swedish sentence to: "Jag tar av strumporna" - and without 'mig' it becomes ambigous, if you mean your own socks or somebody elses, with a slight leaning to somebody elses, since we would prefer to use 'tar av mig' (reflexive verb-form) when referring to 'my own'
There doesn't seem to be a "my" in there in the Swedish - it just means I take off (myself) the socks - they could be anyone's socks that I'm wearing unless I specify "mina" as well as "mig". Plus I'd have thought "strumporna" meant "the socks" not just "I take off socks" - possibly this would work better if the my was included