How and why is Subaru connected to the Witch (Satella)?
Satella The Jealous Witch is in love with Subaru. Why though is not known to the current arc of the latest source arc 6 (web) may change in the official version (Published book).  

The first time we supposed to see her was supposed to be in the current anime arc which is arc 3 it is not shown during the anime but every time Subaru dies he is transported (symbolizes by the arm grabbing him) to a new timeline while that happen he is transported first to a void places (for depiction look at opening 1 where the places is covered with smokes symbolizing the witches miasma which may explain why the smell grew thicker everytime he re-spawn). 

He saw an image of a girl which became clearer every loop. not only thus images pop up the more significant changes that you would observe in the anime (if they didn't cut it) he will hear the whisper of Satella saying 'I Love You' in every attempt of a reveal.(episode 20)

The fact that he say 'I love Emilia' during the Rem scene will put on perspective why Emilia was killed during his attempt to reveal the power