Most of the comments on doing good in secret were examined in "† 11. On Almsgiving" and are applicable here, as well.
Regarding fasting generally, please see the "Appendix: Fasting".
Prefatory Note
‡ Matthew 6:16–18
Matthew 6:16
"And when you fast, do not look gloomy like the hypocrites, for they disfigure their faces that their fasting may be seen by others. Truly, I say to you, they have received their reward.
like the hypocrites
The word "hypocrite" is also translated "actor," and Jesus points out that we sometimes draw attention to ourselves while doing something that is supposed to be private and personal, putting ourselves on display for the world to see. While some in His day evidently went so far as to disfigure their faces, we might today see a milder indicator—but an indicator all the same. If we complain about the effort we make when caring for the sick, feeding the hungry, or doing any other good works, we rob ourselves of the credit God may otherwise have given us for doing that good work.
We ought to do our good cheerfully, shining as lights in the world.
1 Peter 4:9
Show hospitality to one another without grumbling.
2 Corinthians 9:7
Each one must give as he has decided in his heart, not reluctantly or under compulsion, for God loves a cheerful giver.
Philippians 2:13–16
It is God who works in you, both to will and to work for his good pleasure. Do all things without grumbling or disputing, that you may be blameless and innocent, children of God without blemish in the midst of a crooked and twisted generation, among whom you shine as lights in the world, holding fast to the word of life....
Matthew 6:17
But when you fast, anoint your head and wash your face,
anoint your head and wash your face
In other words, make yourself presentable. Don't make it look like it's costing you, or you will draw attention to yourself, losing your reward.
Matthew 6:18
that your fasting may not be seen by others but by your Father who is in secret. And your Father who sees in secret will reward you.