Three general approaches, assuming the cut scenes are videos and not some kind of in game animation.
1) Run in emulator or use model with video out, capture accordingly.
1a) Do bear in mind you can play hacker for this and if the game (or another using the same format) has an intro scene you might well be able to overwrite the scene you want with any and all others up to the super secret new game plus mode end bonus scene or whatever. This being basic file hacking wherein you don't need to know anything about the file format, only what files they are (the game possibly even giving you nice names/extensions/sizes/folder names to look at for this one).
2) Open the game up and hope someone before you has decoded the file format. Games often use custom graphics formats (albeit ones they buy in) as patents (*spits*) and other fees mean it is not just a matter of stick an MP4 in there.
https://wiki.multimedia.cx/index.php?title=Category:Game_Formats for some examples.
Can get marginally better on PC. Console stuff is often quite custom.
3) You figure out the format yourself.
http://www0.cs.ucl.ac.uk/teaching/GZ05/ and perhaps more specifically
http://www0.cs.ucl.ac.uk/teaching/GZ05/09-mpeg.pdf is a light intro to video encoding setups, games for the PSP are not going to be too radical compared to modern PC video playback methods but it is still going to be very involved compared to most other aspects of ROM hacking. Or if you prefer for anything more advanced than motion jpeg (basically a JPG slideshow) then most of anything we have from games is because it is a common format used elsewhere (the patents thing noted above means a fair few games have headed into the Ogg/Vorbis/Theora open source world), the devs of format provided a decoder (
http://www.radgametools.com/bnkdown.htm , though closed nature of things has prevented several games from going open source despite desires from the devs) or someone leaked some source code/players for it all.