The big hand shows the number of minutes, ranging from zero to 59, past the current hour.  So when it is straight up, it means "on the hour." straight down means "30 minutes (half an hour) past the hour. Every minute is one sixtieth on the full circle around the clock, or six degrees of angle.

When we first started making clocks of this type ("escapement clocks" because the ticking mechanism is called an "escapement") a few centuries ago, our clocks did not have minute hands, primarily because we couldn't make them precise enough.  These early clocks had escapements that were driven by gravity - weights on ropes that put tension on the escapement.  Problem was, the weights dropped a little on each cycle ("tick") of the mechanism.  So someone had to go out periodically and pull the other end of that rope to hoist the weights back up.  To allow for more time between hoistings, we put the clocks up high, in towers.

As the mechanisms became lighter and more precise, we added a minute hand...the "big" hand.  That was less then 500 years ago!  How time flies!

If you take the big hand off of a clock, you can still read the time of day, but the accuracy of your reading could be off by 5 to 15 minutes.  With the minute hand, the reading is accurate to typically half a minute. You can imagine the time when, not knowing the time of day more accurately than 15 minutes, things must not have been very hectic.  The minute hand raised expectations about punctuality.  And then the "second" hand much later raised them again.