tsop348 infrared receiver
The BURST cycle

Is there to help the receiver acknowledge there is a signal and synchronize to it so the data is properly framed. In other words give it enough time to set itself up properly. There could be other IR senders with different carriers so it needs to be able to discriminate. Hence the specification to ensure the detector works correctly and doesn't respond to a false signal.

I think you missed a very important part of the drawing - the two vertical ticks on the X axis - this means a BREAK in the time line. 


The diagram was focused on the important part - the data and gives representative timings rather than the full signal which would have made the diagram very wide and less clear. 

Carrier frequency

The LED is being switched ON and OFF with a carrier frequency (usually in the 10's of kHz - in your case 30 kHz) 

The data stream - the bits that carry the information - modulate these 'carrier' pulses of infra red by switching them ON and OFF at a much lower frequency.

The emission frequency of the LED has nothing to do with it EXCEPT it is chosen to be infra-red to try and avoid interference in the sensor with other light sources.