This is known as orbit determination.

https://en.wikipedia.org/wiki/Orbit_determination

A telescope can observe what direction something is in, but cannot directly measure the distance. But because an object's motion depends on its distance from other masses (mainly the sun), three observations at different times are sufficient to determine the orbit. Newton worked that out for the special case of a parabolic trajectory and later scientists refined the mathematics and extended it to all orbits.

The observations have error bars, which means the orbit determination also has error bars. Observations over a short period of time tend to result in less precise orbital information than observations spaced further apart, and extra observations are good to have. (And in some cases, we actually have radar distance measurements.) The error bars on the orbital information mean that a prediction of the future position of the comet is less precise the further into the future it is, and if the comet will pass a planet close that amplifies the errors. For small bodies like comets and asteroids non-gravitational effects, mainly relating to solar radiation, can perturb the orbit and they are hard to predict. All these factors mean don't usually know for certain that an asteroid or comet will hit or miss Earth, but instead you read things like a "1 in 100 chance of impact in 20xx".