The tropopause marks a boundary between the troposphere and the stratosphere. It's significance is that it marks the point where a relatively linear decrease in temperature stops ocurring, and temperature (for our purposes) actually begins to increase with altitude from there. The tropopause represents a global atmospheric inversion, which occurs at it's highest altitudes over the equator, and it's lowest altitudes over the poles. It marks the distinct difference between layers of the atmsophere, or airmases.
Most weather occurs in the troposphere. This includes wind flows, which are modified and created by differences in pressure and in turn by differences in temperature.
What we do with airplanes at higher altitudes is determined very much by temperature aloft. A higher than standard temperature reduces our capability to climb, requires more thrust at altitude, means greater fuel consumption to a point. Knowing what to expect and where to expect it gives us some idea of how to plan our performance, our cruising altitudes, and our anticipated fuel burn.
The tropopause is a useful tool in determining weather and tops, and in considering the jet stream. The jet stream is a wind current found in several places around the globe which moves from west to east at considerable speed, and provides greatly increased groundspeed (and therefore efficiency and reduced cost) when traveling eastbound...and reduces performance when traveling west. Knowing the location of the tropopause is an important clue to planning for the most efficient flight. Where differing airmasses meet and wind is present, we also experience a shear, or a difference in strengh or direction of windflow. Anytime there's shear, there's also turbulence (Clear Air Turbulence, or CAT), so having depictions of the location of the jetstream and knowing it's altitudes can help plan for a smoother flight. Smoother flights, in turn, make for happer customers, and less stress on the airframe and engines.
Roughly speaking, the higher an airplane can fly, the more efficient it's engines can operate, and the higher it's true airspeed will be (speaking specifically about turbine engines). The advantage in flying higher and higher is restricted by the temperature at altitude, as well as the weight of the airplane, in addition to winds aloft. There comes a point for each airplane under a given loaded condition when climbing higher and higher is less efficient. Many airplanes don't make it to the tropopause, many do. For the ones that can climb to the tropopause, performance gains are minimal above that point with respect to efficiency...but it does permit getting above the weather and it does permit getting above much of the anticipated clear air turbulence. It's also a gauge to determine when the point of diminishing returns might be reached in climbing higher, as one considers the height of the tropopause; it represents to some degree that point.
Thunderstorm development and most forms of weather stop or are capped by the tropopause. Knowing the height and changes in the tropopause over the course of a route can help plan and account for enroute weather when looking at pressure patterns and forecasts.
The tropopause is just one tool of many, one element, if you will, when looking at the big picture of the weather.