In this case (as opposed to generally) the hypocrites who disfigure [literally, ruin!] their faces may well evoke the original sense of "hypocrites" as actors in the theater, who typically wore large theatrical masks. Fasting typically accompanied grief, often the sorrow of penitence (Neh 1:4-7; 9:1-2; Zech 7:5; Sirach 31:26; Judith 4:9-13). Yet as Joel put it, the true penitent must rend his or her heart and not merely garments (2:13); Isaiah declared that the true fast was to act for justice (Is 58:6-10). Fasting is a time of drawing close to God by demonstrating our commitment to him. Normally coupled with prayer in the New Testament (Acts 9:9; 13:2-3; 14:23; compare Ezra 8:23; Neh 1:4), biblical fasting is not asceticism for asceticism's sake (Col 2:18-23). Many Pharisees may have fasted twice a week as a mark of piety (Lk 18:12; b. Ta`anit 12a); but I fear that some early Christians missed the point of this passage when they insisted that believers should not fast on Mondays and Thursdays like the "hypocrites," but rather on Wednesdays and Fridays (Did. 8:1).
Under normal circumstances people trimmed beards or changed clothes before appearing in public, as well as anointing themselves. (Palestinian Jews used oil to clean and anoint their skin, especially on their heads; t. Sebi`it 6:9; ARN 3A, probably to lubricate dry scalps.) Because penitent fasting included afflicting oneself (Lev 23:32), for most Jewish people the most extreme fasts meant not only abstaining from food but also practicing other forms of self-abasement like not shaving, washing one's clothes, anointing or having intercourse (m. Ta`anit 1:6; 4:7; Yoma 8:1). Jesus is so concerned with keeping one's righteousness private that he prohibits customary features of what his contemporaries considered a strict fast.
It may be difficult for a member of a family to get around explaining why he or she is not sharing a meal, but in normal circumstances we may wish to observe Jesus' warning as literally as possible to guard our own motives before God. If we want our credit with God, we need to be satisfied that he alone knows, for we can trust that his reward will be more than adequate.