。
She would rather stay at home than go out.
Literal
She [topic-は] going-out [than-より] home [at-に] be-want.
~より ('than / from') marks the standard of comparison — what's being compared *against*. The structure XよりY(の方が) means 'Y more than X.' Here X is 外出する ('going out') and Y is 家にいる ('staying home'); the latter is what she prefers. ~たがる ('want to,' third-person observable form) attaches to 居る ('be / exist [animate]'), giving 家にいたがる ('show signs of wanting to be at home'). The ~がる suffix marks externally observable desire — Japanese conventionally avoids asserting someone else's inner state directly, so when reporting another person's wanting, you use ~がる to say only what you can see. The whole construction packages a clear preference into one tight clause: not 'going out,' but 'at home' is what she's drawn to.