"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."