"Please do not stop walking every day."
の is a verb normalizer. Unlike in English, Japanese can't use a verb (する here) as a subject or object to another verb (やめでください here). That's why の is used. の turns the first part of the sentence (walking every day) into a noun, so it can be used as a direct object.
こと「事」 is another verb normalizer (which means 'thing'), so using こと here would make the sentence literally: "Please do not stop the (walking every day) thing".
For extra information, any noun can be used here. So for example: しりません人 (a don't know person) would mean: "a person (I) don't know". This is different to 人をしりません, which means "(I) don't know the person."
I guess because "aruku" would imply only walking any amount (which everybody does unless they're physically disabled or injured) whereas "sanpo suru" implies a whole Walk, outdoors, which the English sentence probably entails (though it's not unambiguous).
In an another example it was said that する+の was unnecessary, because するturns the noun to verb then の turns it back to noun. Why we need this here?
I guess it's like "renshyu suru" Renshyu is practice when on its own where as when it's with suru it becomes the verb "to practise". Sanpo suru becomes "to take a walk".
I'm interested in why 「のを」was used instead of 「のが」, which is more familiar to me.
やめる is transitive and takes an object naming the thing that is being quit (in this case, the walks). If you used the subject particle が instead of を, it would mean the walks themselves were quitting something.