The people of the Old Testament were redeemed  exactly the same way that people are redeemed today.  By faith.  Either God will do what He says He will do, or He won't.  And you either believe Him, or you don't.  "Believing in God" are the operative words.

That said, to trust someone, you must know their nature.  You must know that they are dependable.  You must know that you can count on them.  And the purpose of the stories in the Old Testament are, in large part, stories to show us how people learned those lessons the hard way.  They lead us to see,  they teach us, through history, who God is and the content of His character.  And whether or not we can trust Him.  Whether He is dependable, or not.

The people would rebel, turn to their natural selfish, rebellious, sinful ways--against which they had been warned repeatedly by the God who created them.  He would give them slack, try to work with them to turn them around, show them mercy.  But eventually, punishment would come.  They would be miserable for a spell, then repent, return to God--and the cycle would start all over again.   Human nature hasn't changed.  We do the same thing today.

The concept of a sacrifice began in the garden of Eden (Read the second chapter of Genesis) when God covered Adam and Eve with skins--after they disobeyed.  They were naked and had no concept of nakedness before then.  He sacrificed an animal.  (Up to that point, man had been a vegetarian.) God shed blood as a covering for the sin that they had committed.

Sin is always disobedience.  The Bible even tells us in a number of places that if you don't know the "rule" then you can't break it.  Only when we knowingly go against God do we disobey.  And the shedding of blood to cover our rebellion against God prevails in those books of the OT.  

So God required repentance  from people of old just as he does today.  And their ceremony of repentance involved a sacrifice.  They had to give up something to seal the agreement.  And back then, it only lasted a year.  Then they had to do it again.  And again.  For the rest of their lives.