Just so we know what we’re talking about:
Slaves, obey your earthly masters in everything; and do it, not only when their eye is on you and to curry their favor, but with sincerity of heart and reverence for the Lord. (Colossians 3:22)
Slaves, obey your earthly masters with respect and fear, and with sincerity of heart, just as you would obey Christ. (Ephesians 6:5)
As the question is asked, we are supposed to look at these verses on their own (that is, out of context). Reading them together, however, one might be argue that they are supposed to be about humility and sincerity. If you are going to do something, do it with all of your heart. Do not act simply for gain.
Putting the verses in context, on the other hand, one might look at the whole of Colossians 3 or Ephesians 6. Consider the following passage, for instance, which comes shortly after one of the verses with which this question is concerned:
And masters, treat your slaves in the same way. Do not threaten them, since you know that he who is both their Master and yours is in heaven, and there is no favoritism with him. (Ephesians 6:9)
If we look at the chapters in whole, they are clearly about duties that we all have to one another. They tell people that their earthly stations are temporary, and that they had better fulfill them with humility and in full knowledge that they will be judged in light of what they made of the lot they were given in life. The passages do not denounce slavery, and we may very well find that a failing of these letters (or of their author, Paul). Still, we should not pretend that these passages are endorsements of slavery either.
We might think that these pronouncements about the proper relationships that members of a household have to one another are outdated, and I certainly would agree, but that is a slightly different issue. We might also be horrified that Paul was content to treat slavery as just a fact of life without commenting upon its moral indefensibility. I’m not here to say that these are not legitimate reactions. It is entirely possible to read these passages as being about personal virtue and integrity, however, and not as commitments to the continuation of a particular status quo.
I am not a Christian, of course, and I cannot speak for Christianity. I am simply giving what I think is one possible way of explaining these passages. It is also worth noting, as @Aethelflaed has already, that Colossians and Ephesians are both in the New Testament.