Eagle Schematic help for 74AHCT125 quad level shifter
The 74HCT125 is offically a quad (4 channel) bus buffer, not a "Quad Level Shifter", as Adafruit calls it.

Your "transistors" are the common schematic symbol for a tri-state buffer - a digital logic gate.  for multi-channel parts such as this, it is common to show sections individually on a schematic, rather than one large block containing all the channels.  That makes it much easier to understand the schematic without having to go find the datasheet to see what's inside the big box.

the 74HCT125 has four separate buffers, each with three terminals (in, out, and enable) as well as power and ground connections common to all buffers.

For IC1A, pin 2 is input, 3 is output, and 1 is enable.  You should get the datasheet for the 74HCT125 to learn more about the chip.  There is a link to the datasheet on the Adafruit page you linked to.  (Don't just go by Adafruit's description)