Why would you use a subjunctive in this sentence? "Quiero que venga conmigo."

The subjunctive refers to a mood that deals with any form of uncertainty, as opposed to the indicative which deals with certainty and factual actions. "Quiero que venga conmigo" expresses a desire which represents a form of uncertainty. Therefore the subjunctive is used rather than the indicative form. The key thing to remember here is that the desire is about another person. This is what introduces the uncertainty because when we talk about our own desires the indicative tense can often be used. For example, if someone were hungry they might say: "Quiero comer algo". This sentence does not involve another person and so the subjunctive is not required.