How do Mitzvot bring us closer to God?
Obedience to God does not bring us closer to him.
Obedience is the expression of recognizing authority, and authority is an expression of distance. The more distant a person is over us, the more authority they have over us, which makes us more obedient. There’s the obedience we have to the regional manager, and then the obedience to the CEO of the company. Obedience means distance, not closeness. You don’t feel closeness with the person who obeys you or vice versa.
Closeness can only come when we are invited by the person we want to get close to, to participate in something that’s already close to them. When there is something close and precious to them, and they invite us to participate with them in that thing, that brings closeness. If there is nothing that is already precious to them, there is no way to get close. We become close when we get involved with something closer to them than ourselves.
When God tells us to keep Shabbat, its not a commandment, its an invitation to participate in something that’s already precious to God. God says “My Shabbat,” or “My Yom Kippur,” or “My Friday Night Candles is very important to me, please do it with me.”
That is how we become close to him.
This is why the word Mitzvah doesn’t really mean “a commandment,” it means “a connection.”