First of all, an enzyme is a protein. Taber's Cyclopedic Medical Dictionary defines an enzyme as follows: "Enzymes are complex proteins capable of inducing chemical changes in other substances without being changed themselves."

How do Enzymes function?
Enzymes are the workhorses of the body. Enzymes break down carbohydrates (sugar and starch) to provide energy; they are intimately involved with muscle contraction, nerve conduction, digestion and practically everything else that happens in the body; without enzymes we cannot not live.
Some enzymes break things apart, some change the shape of molecules and others join molecules together. Hydrolytic Enzymes cause a chemical decomposition in which a substance is split into simpler compounds in the presence of water.
Below are some examples:

Amylolytic Enzymes - the hydrolysis of starch into dextrins and/or sugar

Amylase enzymes + Starch + H2O --> Maltose (a disaccharide sugar)

Lipolytic Enzymes - the hydrolysis of fats into glycerol and fatty acids.

Lipase enzymes + Fat + H2O --> Glycerol and Fatty Acids

Proteolytic Enzymes - the hydrolysis of proteins into peptides and/or amino acids

Protease enzymes + Protein + H2O --> Peptides and/or Amino Acids

Yes, enzymes are absolutely safe. Decades of research was conducted in Japan and Germany during the post World War II period. Prisoners were clinically tested with enzymes to study their toxicity. To their surprise, researchers found that even when given doses equivalent to 3000 tablets of enzymes through the renal route (which has a very high absorption rate), the maximum adverse effect that was observed was Diarrhea.

Due to such low & negligible toxicity, enzymes are preferred by all the alternative medicine practitioners and naturopaths worldwide. In fact, in countries such as United States, Germany & Japan, it is uncommon to find people consuming 15 - 30 capsules/tablets daily for their healthcare needs.

Conventional drug therapy, although giving some desired beneficial effects, also ends up giving undesirable and mostly unavoidable side-effects.
Enzymes, offer a serious alternative therapy to consumers and practitioners, who are tired of side-effect-prone drugs and who are looking forward to a much more fulfilling lifestyle.

Systemic enzymes are those enzymes which have the ability to have a system wide effect through-out the entire body. They are formulated in such a manner so that they are able to withstand the acidic conditions of the stomach and reach the intestine. They get absorbed, in the intestine and ultimately reach the blood where they achieve systemic circulation. Systemic enzymes are usually proteolytic enzymes.

Digestive enzymes, on the other hand, have a broad range of enzymes for protein digestion (proteases), as well as for the digestion of fiber (cellulase), starches (amylase) and fats (lipase). The actions of the digestive enzymes are limited to the GI (gastro-intestinal) tract. After being used for digestion, digestive enzymes generally do not reach the intestine, either due to insufficient quantities or due to lack of protection.
People using digestive enzymes thinking to use them as systemic ones have failed because of the lack of protection of these enzymes against the stomach acid and lack of balance between the components themselves. Systemic enzymes need to be tweaked and formulated to ensure synergy of action. They also need to be put into "suspended animation" so as to keep them from eating each other & thus neutralizing their own action.