When the first tanks rolled on the scene in WWI, there was little the lone infantryman could do other than run; which wasn't to hard since the Mark I had a top speed of 4 mph.
 
But eventually the tanks got faster, and infantry needed some options to deal with them. First on the scene were anti tank rifles but increasing armor on tanks by mid-war made them ineffective. Until the invention of shaped charge explosives anti tank weaponry required high velocity, and large explosive warheads, to penetrate armor. These rounds could only be fired from large guns and/or were too heavy for infantry. Although some of the first shaped charge weapons were grenades, bombs, and mines, infantry desperately needed a weapon which allowed for some stand-off distance... Enter the panzerfaust and "bazooka".
 
In Axis and Allies RTS the anti tank infantry regiment has a limited role, but when it is deployed, it's usually by Japan and Germany.
 

The primary benefit anti-tank infantry offer is the ability to paralyze enemy vehicles, which allows extra time to kill a retreating enemy.
 
Anti tank infantry will not matchup well versus most infantry, especially sniper-based. Scout your enemies to avoid a matchup problem.
 
Anti tank infantry works best when used in coordination with mech or armor, as they are often too weak to fight on their own, and too slow to finish off retreating vehicles.
 
Infantry is at a disadvantage versus mech and armor. Very important to keep your anti tank infantry in benefitial terrain to help level the playing field.

Late in the 1800's Charles E. Munroe discovered the "Munroe effect", that a void or hollow area in an explosive charge could be used to focus the energy of the blast onto the target. In the early 1900's the Germans refined the idea and realized an explosive charge with a cone shaped void could punch holes through plates of steel.
 
It wasn't until WW2 that these discoveries found a military application, and for the first time a single infantry soldier could destroy a tank; a fact which has affected tank design to this day. The Panzerfaust HEAT warhead is a good example of how a shaped charge warhead works, because focusing the explosion is only part of what makes it effective. You must also detonate the explosion at a precise distance from the armor plating of the target. Like starting a fire with a magnifying glass, if you are too far or too close to the target you will not achieve the desired effect. Most shaped charge weapons achieve this "stand-off" with a hollow area at the tip of the warhead, such as the cap of the Panzerfaust HEAT above.
 
Some Russian tank crews used improvised cage/slat armor by welding anything they could find onto tanks to protect from shaped charge attacks, such as metal furniture and bed matresses. The Germans came up with Schurzen, which was armored skirting, often covering the tracks and sides of the turret; which achieved the same effect. If you could detonate the warhead before it contacted the main armor you could survive most attacks, although in modern times some anti tank weapons have 2-3 warheads made to punch through multiple layers of armor.
 
Infantry based anti-tank weapons had a huge effect on the evolution of the tank, underscoring the need for mobility, as well as armor and firepower. Most of the super-powers had super-heavy tank designs in various states of completion, such as the Maus, O-I Ultra Heavy, and the TOG2 but were abandoned due to the evolving tactics of warfare.

If you are going to deploy a large number of AT infantry, you better get every damn upgrade possible.... See: Upgrading Your Infantry/Airborne

Can't get enough stats? Here's a breakdown of the Anti Tank Infantry regiment by unit, before/after upgrades. Click on the icons to sort the columns

Here's our unofficial ranks, by unit and regiment for Anti Tank Infantry. Click on the icons to sort the columns