People specifically need to STAY on about their business; otherwise, how would they notice what is "amiss" or "out of the ordinary".
I used to think this question had only to do with a person's sense of "security" while traveling domestically or abroad. It could be this. Nonetheless, when one is traveling (vacation or business) to a "not usual" locale, one thinks (he) they wouldn't notice anything unusual and because everything there is "unusual" from the standpoint of a guest. Not so. When one is away, one's "security register" is in higher gear, usually, than when one is in one's routine daily life. Therefore, conducting business/life as usual is more than just an operative phrase and directive.