There isnt really much to understand about this game. Pretty much you're the sole human being in a distant future and you travel around time getting time crystals to stop the timesplitters (the main antagonists). And no, you don't really need to play the first one, the levels aren't really interconnected.
"I attacked Vivec and my game crashed" "Obviously the demigod's last line of defence." -AnilEhilated
I believe there is -no- connection to TS1 what-so-ever except in concept. However, the sequel to this game is a direct sequel, picking up where this one leaves off (not that it's important, this one's story is pretty minimal, and self-contained).
More important to remember about this game is that each level has a pretty stand-alone story, that hardly interconnect except for the 'timesplitters.'