Native Structures are Compact
One of the important factors for humans or computers trying to fold proteins is achieving compactness.
A newly-formed string of amino acids in a cell has a preferred three-dimensional shape (its “native structure”) and the components in the interior of the folded molecule like to be snugged up together tightly, without spaces between them. In Foldit, you can get some idea of whether your protein has such spaces, called voids, by selecting “Show Voids” from the view options menu. Few players use this view to fold, but it's useful checking the quality of a fold.
Seeking the Low Energy State
That’s the short story, but why do proteins like to be compact? It’s a story of love and hate - of opposing forces. First, some of the sidechains on the amino acids are hydrophilic (water-loving) because they carry slight electric charges, which can form weak hydrogen bonds with water. The amino acid backbone itself also is weakly charged and prefers to be in contact with water. In the other camp are the hydrophobic sidechains, who will do their darnedest to avoid contact with water. These forces of attraction and repulsion will seek equilibrium in the cell’s watery interior. So after a protein is synthesized, it starts to collapse such that the hydrophobic sidechains form a core, and the hydrophilic sidechains face outward, shielding the core from water contact.
The conformation that best achieves this balance of opposing forces in a kinetically stable way is termed the “low energy state” for that protein, and it corresponds to the protein’s native structure. Compactness for its own sake is not really the goal, rather it is a by-product of the hydrophobes trying (but not all succeeding) to barricade themselves tightly inside the protein against their sworn enemy, water. Chemists term this “hydrophobic interaction” but there’s no bonding or strong attraction going on. It’s merely avoidance behaviour.
Sheets and Helices Form in the Core of the Protein
There are other forces at play when the protein folds, like the attraction between charged sidechains, the angle of the backbone bonds, etc., but the main issue is hydrophobic hiding. However, the hydrophobic sidechains are of course attached to the backbone, which itself is charged -- each amino acid has both a positive and a negative side (it’s “polar”). So how does this polar backbone remain stable when part of it is tucked into the center of the protein, a non-polar environment? It manages rather elegantly. The backbone forms itself into helices and sheets, which can form hydrogen bonds with themselves, thus neutralizing the charges, and allowing peaceful co-existence of the backbone with its non-polar (hydrophobic) indoor neighbors. If a sheet is always parallel at least with one another, helices are often in orthogonal bundles.
There are some proteins that have holes in the interior that are filled with one or more water molecules in their native state. In this case, the water is H-bonded to polar entities on the interior, and the water is considered an integral part of the protein structure.
What Do Sheets and Helices have to do with Compactness?
This internal hydrogen bonding of the backbone is another reason proteins tend to be compact – helices and sheets are by their nature compact structures and are very stable even within the hydrophobic core, so their formation is favored during protein folding.