Meat is cut against the grain in order to improve the eating experience. In working beef primals like the flank (pictured below) the muscle fibers are clearly defined.
By cutting the meat against - more specifically - perpendicular (90 degrees) to the grain you effectively shorten the muscle fiber lengths and make it easier for your teeth to separate the meat.
In order to better understand what the grain is, it's important to have an understanding of muscle anatomy and composition.
Muscle tissues are comprised of bundles of cells called muscle fibers or fascicles. Muscle fibers are grouped into muscle bundles composed of myofibrils. The myofibrils are comprised of proteins actin, myosin, and others - referred to as myofibrillar proteins (myofilaments). This is the general muscle structure for beef, poultry, pork, and lamb.
Essentially, muscle fibers are long strands and the direction in which they form is the grain.
This is in a similar vein to the wood or lumber industry, "Grain" describes the growth pattern of the tree.
On working muscles, the grain direction is more pronounced as the muscle fiber bundles are thicker.
Pictured below is a piece of flank steak. The grain structure or muscle fibers are very pronounced which makes it easy to determine the direction to slice.
Here's another example of a rump roast. Again, the grain structure is clearly defined.
Another example of skirt steak:
As you can imagine, this is sort of problematic on smaller, weak muscles like a loin or tenderloin where the muscle fibers are very fine. Granted, the muscle fibers are still present, only they're harder to identify.
In these cases, regardless of the direction you slice, the bite will be soft and tender regardless. If you're the person who is preparing the meat, grain is more apparent on raw meat than it is on cooked meat.
The overarching goal is to shorten the muscle fibers as much as possible in order to make chewing easier.
The next time you have a piece of meat like a flank steak or rump roast, slice a piece of meat both with and against the grain.
Attempt to rip the muscle fibers with the grain - It's almost impossible.
Attempt to rip or tear the muscle fibers against the grain - much easier.
America's Test kitchen did a quantitative analysis of this which helps to further illustrate the amount of force needed to "chew" both with and against the grain. Their test also proves my point in regards to weaker muscles grain direction not mattering as much.
They used a CT3 texture analyser in order to simulate how much force is required to cut (bite) 5 mm into the meat.
Force Needed to Bite
|Flank Steak||Strip Loin|
|with the grain||1729g||590g|
|against the grain||383g||329g|
On a working muscle like the flank, you need 4.5x as much force. Where-as on the strip loin you need 1.79x as much force.
Technically through trigonometry and knowing that you need to cut perpendicular to the grain (90 degrees) you could minimize the length of the meat fibers (making the length of the meat fibers the same as the width of the slice).
However, this is hardly practical and not many kitchens make use of a protractor to cut their meat.
On a working muscle, the result will be a tougher/chewier piece of meat. For instance, with steak, if you were to attempt to chew with the grain, you'd need to use significantly more force which results in a markedly chewier piece of meat.
Pictured above is a rump roast sliced both with and against the grain. Again, you can sort of see what I mean by if you were to tear the meat with either your hands or teeth; The chew is made easier when the muscle fibers are shorter - against the grain.
Note: The pieces that were cut with the grain were fed to my dogs as the above meat was being used for jerky.
This is one of the most common mistakes in regards to meat grain. Sear and/or grill marks have nothing to do with the grain of the meat.
Sear marks are the result of the maillard reaction of amino acids and are independent of meat's grain structure.
A good portion of the world consumes chicken - people also seem to overwhelmingly love white meat over dark meat. In a similar vein to beef, slice on a bias perpendicular to the grain.
Chicken breast's typical grain structure looks like the following:
Again, you do not need to be super anal in terms of exact cutting angle. Doing so with chicken breast is near impossible as the grain/striations tend to curve. However, slicing against the grain on a bias is still better than slicing parallel to the grain for much the same reason as beef.
The overarching reason it's recommended to slice against the grain is to make the meat more tender and to promote a better eating experience.
Meat that is sliced parallel to the grain will result in a chewy mouthfeel and require more effort to break-down. Muscle fibers that are cut against the grain are much easier for the teeth to separate.