In the block diagram of the internal circuit, notice that when transistor Q5 conducts, OUTPUT is connected to ground. But if Q5 does not conduct, then OUTPUT is not connected to anything, leaving its voltage level indeterminate.
By connecting OUTPUT via a resistor to a voltage source, it now always is connected to something, and the only question is which connection determines OUTPUT's voltage. With the addition of the pull-up resistor, when Q5 does not conduct there is a zero-current connection between OUTPUT and the connected voltage source, causing OUTPUT to have the same voltage as the source. When Q5 does conduct, OUTPUT once again is connected to ground, even though some current flows through the pull-up resistor. (To keep the heat dissipation in the resistor low, you would pick a high resistance. 1 kΩ or 10 kΩ are common choices for a pull-up resistor.)
It is very common for integrated circuits to be designed to require the use of a pull-up resistor. Doing this permits multiple devices of this kind to be connected to the same signal (OUTPUT in this case) while using just a single pull-up resistor. The effect is that any one of the integrated circuits can pull OUTPUT low. The only way for OUTPUT to be high then is for all the integrated circuits to agree that it should be high. This is often called a wired-AND: the output is high only if all inputs are high.
In computers, a common use of this approach is to permit any one of a number of devices to pull a line low to signal an interrupt. Of course, then the computer still has to determine which device did so by some other means. Seeing that the signal is low does not suffice to identify the device that made it so.
The other approach to the output stage of an integrated circuit is generally referred to as a totem-pole output. In such a circuit, one internal transistor can be activated to pull the output low—as in the circuit you presented—and another one can be activated to pull it high. If neither output transistor conducts, then the output is not controlled. It is said to present a high-impedance or high-Z output, a fancy way of saying it is not connected. With such an output stage, you might omit a pull-up resistor altogether, if it is acceptable for the line to have an indeterminate value. (If it is not acceptable, you might as well use an open-collector-output or open-drain-output integrated circuit, like the one you present.) Such integrated circuits can be used to control a single line if you can guarantee that one and only one integrated circuit ever will control the signal at one time. If you violate this restriction, you will create short circuits that might damage devices.