On lighting a gas stove the flame is active only above the burner. However, the gas input from beneath the burner and all the way to the pipeline contains gas fuel. Why doesn't it all ignite up ? I am guessing it may be due to the pressurised flow of the fuel in one direction but then I am unable to relate it to combustion, i.e., if a combustible substance is in contact of a source of ignition then it should start to burn.
The methane that goes to the stove contains little or no oxygen, so it will not burn until oxygen gets mixed into it. This occurs at the burner to the extent that a blue flame is formed in order to obtain complete combustion. At the point where air is mixed into the methane, the velocity of the flowing methane and air is kept high enough to exceed the methane flame velocity, so the flame cannot go backwards through the burner. In addition, the metal parts of the burner are substantially colder than flame temperature, and if the gas flow is decreased to the point where the flame tries to go backwards through the burner parts, the close clearances and cold metal temperature of the metal burner parts quenches the flame.
A good example of the type of behavior described above can be seen when a bunsen burner is lit. The chimney of the bunsen burner is adjusted such that the air register at the bottom of the burner admits enough air to obtain a blue flame. If you continue opening that air register, the air and gas flowing up the bunsen burner tube increase in velocity until the flame has difficulty burning fast enough to keep up, and eventually, the high velocity up the burner tube blows the flame away from the burner, and the flame goes out. On the other hand, if the gas and air flow are decreased to a very minimal amount, the flame cannot propagate down the tube because the tube diameter is small, and the flame gets quenched by the "cold" walls of the bunsen burner tube.