A person carefully filters a drink to remove a gnat and then somehow swallows a camel. Jesus' image is intentionally absurd, and its humor makes the criticism memorable.
Both creatures were unclean under the Torah, but their enormous difference in size turns the saying into a picture of badly distorted priorities.
Small insects could fall into liquids, and straining a drink was a practical way to remove unwanted material. For someone concerned with ritual cleanness, inadvertently consuming an unclean creature added another reason for care.
Jesus then leaps from one of the smallest imaginable contaminants to a camel, an enormous unclean animal. The saying works as hyperbole; it does not describe an actual practice.
The background preserves the joke as well as the rebuke. The hearer pictures meticulous avoidance of the tiny violation while an impossibly large one passes unnoticed.
Placed after Jesus' criticism of misplaced religious priorities, the image warns that scrupulous attention to small matters can coexist with blindness to much larger failures.