Killing L killed Rem because L was close to catching Misa again and she could have been executed. So to answer your question, it's both love and increasing a lifespan. RIP Watari though. She killed him for no reason.

Did REM really kill L?

Rem, being pretty sharp herself, came to the same conclusion. She knew L would figure it out, she knew it would mean the end for Misa. Though, she had made a promise to protect her till the end, so she knew she had to kill L to prevent Misa from dying early.

Does L Really Die?

Twenty-three days after writing his name in the Death Note, and after burning all the remaining Death Notes and having a conversation with Soichiro Yagami, L dies peacefully while eating a chocolate bar, with a picture of Watari lying right beside him.