Because it shows time using definite digits, rather than positions of hands that have to be measured. (It would be less correct to say "because it has a computer inside.")
Binary means "two states." The two states are sometimes called "1" and "0", or called "true" and "false", or called "on" and "off", (or other names.) The essential characteristic is that a single binary device can be in just one of two possible states.
A bit is a single "on/off" value.
A good example is a toggle switch, such as a light switch. You can turn it "on" or "off" but not (in normal operation) anything else. A light switch holds one bit of information.
A light dimmer is not a binary device: it has many positions between "off" and "fully on". If you want a light dimmer to be set to 25%, you must carefully adjust it.
Which of the following is a binary device: