Consider this scenario:
Imagine a store that is open for a certain time interval during the day. In that time interval, several customers rush in to the store. Each customer buys several items. If someone would record the number of items sold as a function of time, he/she would probably observe bursts (one burst = items bought by one person). The time duration for which the store is open will correspond to the number of bursts. In other words, time duration maps to frequency of bursts.
Now, consider this transcription model:
Consider this mechanistic/intuitive model explaining how proteins are produced in bursts and how the frequency of bursts are controlled by the transcription factor:
1. transcription factor binds to promoter regions and opens the region for access by the polymerase
2. the region remains "open" for some time
3. during this time interval, the polymerase may initiate transcription multiple times
4. for each mRNA that the polymerase transcribes, multiple proteins are produced
So, in summary, the transcription factor "opens" the promoter region for the polymerase. Lets assume that upregulating the transcription factor affects the time duration of the "open" promoter. The longer the time interval, the more mRNA will be produced. Each mRNA creates a burst of proteins. Therefore, upregulating transcription factor affects frequency of bursts.