He maketh me to lie down in green pastures: he leadeth me beside the still waters.
Here the metaphor of sheep and shepherd is still goin' strong. The speaker is declaring himself a sheep here, so don't be surprised if he acts like one.
The Lord takes the speaker to fertile green pastures where he can eat his fill and then lie down to rest. This is a good image because green pastures are alluring to both sheep and humans, though for different reasons.
We may think that a green pasture is just a beautiful, peaceful spot, but for a sheep it's a vital necessity in order to find enough food. This line, then, gives a specific example to back up the point that the shepherd won't let his sheep lack any essentials.
The Lord also takes the speaker-as-sheep to calm waters to drink and refresh himself. His thirst is quenched: another example of a necessity met.
The availability of food and drink also serves as a nice implied metaphor for the way that God satisfies the speaker's spiritual (as opposed to literal) hunger and thirst.
The stillness of the waters indicates that there are no storms or winds to disturb this peaceful scene. Also, a still pool of water is probably more pleasant to drink from than some rushing stream. We can vouch for that one.
The word "maketh" does not mean "forced" here – don't picture some shepherd shoving a poor sheep down into the grass! It means "creates the opportunity for."
Don't be put off by all the old-fashioned diction. That's just the King James Version for you.