What is the effect of gradual increase of speech speed on speech comprehension? When I am watching a YouTube video with normal speed I can understand everything with no problem. When I speed up the video to 1.25x my understanding of what someone is saying is a bit worse, but after 1-2 minutes it is very clear again.

I have made an experiment in which I was increasing the video speed gradually to 2x (in ~5 minutes) and I was understanding everything very good. After some time I started the same video again, but then I switched it directly from 1x to 2x and I still could understand what someone is saying in the video, but it was worse than in gradual speed increase.

Why is this happening?
The ability to understand speech at increased voice rates is all about learning. A notable example in this regard is the use of text-to-speech software like Jaws used by blind folks unable to read the written word. Jaws is able to convert text on screen into the spoken word using a speech generator. After people get used to read on-screen content through speech, they are often able (and happy) to increase the voice rate settings to scan over the (often cluttered) computer screens more quickly. A recommended incremental step for Jaws in speech rate is 5%. I have experienced the speech rates deployed by some blind folks and the resulting speech was totally unintelligible to me. 

With regard to the learning curve you describe: I bet that after repeated exposure to higher speech rate settings you wouldn't feel the necessity to start at a slower pace. Assuming you have normal vision, you are probably not too inclined to practice this for hours, though.