The main purpose of the warm up is to prepare your body for the workout that is going to follow.
It also serves to get your brain ready for the session, so that you are no longer thinking about work, family or whatever other pressures or distractions, and instead you become ready to participate in the session with a clearer mind.
After a long day (or a long week!) at work, you might be quite stiff and reluctant to move. If you launch into too physical or too demanding a workout straight away then you may cause yourself an injury, so you need to ease yourself into it. Get your blood moving, loosen off your tight muscles, get your joints to move without creaking too badly – prepare for the physicality ahead.