What is the Bible talking about when it describes believers earning "rewards" in heaven?

ANSWER: The Bible is clear about the fact that God does give rewards, both in an early sense and an eternal, but we have to be careful in the fact that the Bible never refers to our salvation as a "reward'.” Our salvation is only by faith alone in Christ alone and not(never) by our works.

TITUS 3:4-7 But when the kindness of God our Savior and His love for mankind appeared, He saved us, not on the basis of deeds which we have done in righteousness, but according to His mercy, by the washing of regeneration and renewing by the Holy Spirit, whom He poured out upon us richly through Jesus Christ our Savior, so that being justified by His grace we would be made heirs according to the hope of eternal life.  

EPHESIANS 2:8-10 For by grace you have been saved through faith; and that not of yourselves, it is the gift of God; not as a result of works, so that no one may boast. For we are His workmanship, created in Christ Jesus for good works, which God prepared beforehand so that we would walk in them.

Nevertheless, since God saved us He then calls us to perform good works (Ephesians 2:10). This is the work by which we are rewarded. In fact, God encourages us to work for these rewards as if they are literal “treasures.”

MATTHEW 6:19-21  Do not store up for yourselves treasures on earth, where moth and rust destroy, and where thieves break in and steal. But store up for yourselves treasures in heaven, where neither moth nor rust destroys, and where thieves do not break in or steal; for where your treasure is, there your heart will be also.

Therefore we have to conclude that not only is it Biblical for believers to do good works with an eye toward heavenly rewards, it is also God’s will for us!