The Kill Feed is an element found on the left side of the screen. Any time a player on either team scores a takedown, a notification pops up in the kill feed. These notifications stack on top of one another, and have team-colored visual effects, so it's easy to see exactly what went down during the last team fight.
A player's hero portrait appears on the left side of kill feed notifications when the player scores takedowns, and the heroes who were killed appear on the right. Killing an enemy hero kicks off a killstreak timer. The more takedowns players participate in while the timer is active, the more enemy portraits are added across from theirs in the kill feed, and the more intense their on-screen killstreak callouts become.
Also, if an enemy hero killed has their quest progression reset or partially lost on death (such as Kael'thas' "Convection" or Butcher's "Fresh Meat"), the quest icon is shown at the bottom right corner of their portrait.