What does it mean that believers are unleavened?

Being unleavened means the believer has been forgiven and cleansed, and is, so far as God is concerned, without sin. This pictures the nation of Israel after Passover, whose sins were atoned for, and it is grounded in Christ our Passover being sacrificed.

Les explains that the Passover in Egypt was followed by seven days of unleavened bread, picturing a nation whose sins had been atoned for or cleansed. Paul brings that same analogy into our teaching today, so we must look at sin in our individual lives and in the church, just like the leaven in Egypt's bread. To be unleavened means we have been forgiven, cleansed, and are, so far as God is concerned, without sin. However, if we leave a little leaven, it will expand and consume us. This is because Christ our Passover is sacrificed for us.

1 Corinthians 5:7

Purge out therefore the old leaven, that ye may be a new lump, as ye are unleavened. For even Christ our passover is sacrificed for us:

Why are believers supposed to be an unleavened people?

Believers are to be an unleavened people because their sin problem has been removed through forgiveness and salvation. Paul writes that Christ, our Passover Lamb, was sacrificed for us. Therefore, having been made a new lump, they are to put away the old leaven, which represents evil and sin.