The term “chip” is used as a euphemism in this case. In fact, these devices are a growth of tendrils that wholly replace synaptic regions of the brain.
While an agent has a Brain Chip installed, the corresponding skill level is fixed at 3, and cannot be changed by purchasing skill upgrades. If a Brain Chip augment is removed using an Augment Drill, the corresponding skill level reverts to 1, even if the agent previously had a different skill level before installing the Brain Chip; but they will be able to purchase upgrades for that skill again.
The Brain Chip augments interact strangely with Draco's Neural Pattern Grid augment. When one of Draco's skills is set to 3 by a Brain Chip augment, he can still receive upgrades to that skill by neurally scanning corpses, but the upgrades to that skill will be reverted and his skill level will be set to 3 again at the start of the next mission.
The Brain Chip augments are generally not considered to be good augments, but they can be useful in specific circumstances. The benefit of a Brain Chip is being able to upgrade a skill level cheaper than paying for the upgrade directly, at the cost of an augment socket. The problem is that the skill level is fixed at 3, and cannot be upgraded further.
The Brain Chips can be useful in Campaign modes when playing without the extended campaign, especially on higher difficulty settings when you cannot afford to upgrade all of your skills beyond level 3 on every agent before the final mission. Brain Chip augments can also be given to Sharp in order to get the bonus from his Modular Cybernetic Frame. However, in longer campaigns they are less useful, and in Endless modes they should normally be avoided completely because eventually you can afford to upgrade every skill to the maximum level on every agent.
Perhaps the most effective use of the Brain Chip augments is to carry one or two into the OMNI Mainframe mission, to install on Central or Monst3r. The Kinesiology Brain Chip upgrades them from Speed 1 to Speed 3, helping them get to their objective rooms, and allowing them to use some items like Cloaking Rigs and Stims. The others are less useful, but not useless; the Legerdemain Brain Chip allows Monst3r to use items like Buster Chips or Shock Traps, the Cryptology Brain Chip allows Central to use items like EMP Packs, and the Physiology Brain Chip allows either to carry an additional two items without being encumbered. These options are otherwise not available to Central or Monst3r if you don't start with them, since you cannot purchase skill upgrades for them in the normal way.