Their observation is fun, and if done with care, yields valuable scientific data.
You may wonder why bother with telescopes with their restricted fields when one can look for meteors across large areas of the sky with the excellent panoramic detector, the human eye. This is a good question. There are a number of reasons for studying telescopic meteors. Here are some.
Generally when one observes a telescopic meteor it is the result of a meteor particle (meteoroid) of extremely low mass burning up in the upper atmosphere: the mass may be several orders of magnitude less than meteoroids which cause naked-eye meteors. It is only by studying interplanetary debris over a wide range of the mass spectrum, that we learn the physics of meteor streams. We can draw an analogy with modern astrophysics, where many unexpected and exciting discoveries are made from observations in the other regions of the electromagnetic spectrum, e.g. infra-red and X-rays. Thus photographic, visual, and telescopic observations of meteors are complementary.
The combination of the magnification and restricted apparent field of view lets the telescopic observer plot the paths of meteors more accurately than a visual counterpart.
Plots of meteors have two errors: orientation and displacement. See the diagram. For an average experienced visual plotter the direction has an uncertainity of ±5 degrees, and lateral displacement have errors typically a few degrees. This imprecision of the meteor's path leads to ambiguities assigning a meteor to a shower, especially important for a minor shower. Traced back to the radiant, these uncertainties can amount to several degrees.
Contrast that with the telescopic plotter. First of all, if there were no other factors, magnification reduces the displacement pro rata. So for a typical magnification of 10x would pinpoint the meteor trail to about ±20 arcminutes. However, there's more. A typical viewing diameter for a visual observer is at least 100°. Contrast that with 55° for the telescopic observer with a wide-field eyepiece. That's only 30% of the area. So a higher proportion of meteors are seen directly rather than with averted vision. This markedly improves the plotting errors. Exactly how much is debatable, but a factor of two is certainly not out of the question. This leads to a typical displacement at the radiant of about a degree. It then becomes possible to detect weak minor showers, because pollution by sporadic meteors is reduced. Also probing shower radiants for substructure and measuring radiant motion become feasible.
The plotting of meteors does reduce bias to known showers. Shower assignment is then not a snap irrevocable decision under the stars. Given details of a meteor's speed and its start and end points, and the uncertainties of measurement, a more-objective decision can be made later. Plotted meteor data of several observers or from several years can be combined to detect weak showers, or confirm the presence of a suspected shower. Likewise we can compare or merge data from different techniques: visual, photographic, and video.
Through the eyepiece, there is less bias, because potential radiants lie beyond the field of viewobservers just record the meteors as seen. This is especially the case when using a star diagonal where the lateral inversion and field rotation randomises the bearings to the radiants.
Once you've experienced the thrill of a bright meteor close-up, you'll not forget it. You can see the meteor train form, expand, drift, and decay as if you were only a few kilometres away.