How do I get less stiff with my left hand on the guitar?

You're use to playing an acoustic guitar which uses a heavier gauge string, has a higher action, and depending on what the scale length of the electric guitar is possibly a shorter scale length than the acoustic guitar you’re playing. Because of these elements, you’ve developed the necessary touch to play an acoustic guitar where the fretted notes are articulate. Now you’re playing an instrument which has a lighter string gauge, lower action, and possibly a shorter scale length.

Now back to my initial one word answer “relax.” The difference between playing an acoustic guitar and an electric is touch. With an acoustic you’re pressing down on the string because of all the element I mentioned in preceding paragraph. Where with an electric guitar you really only have to touch the strings. What I mean by that is, you want to just “touch” the strings with enough pressure to get an articulate fretted note. Don’t concern yourself with pressing the string hard enough that the string contacts the fret board. All you want to focus on is the string making enough contact with the surface of the fret which produces a clear articulate note. So “relax” your hand and arm all the way up to your shoulder, and just “touch” the strings don't “press” down on them. If you do this, I believe you’ll be on your way to developing the touch you need for the electric guitar.

One other thing. The same goes for your right hand, you don’t need the same string attack with your pick hand on an electric guitar as you do an acoustic. The funny thing is the harder the pick attack with your right hand, the harder you press down on the strings with your fretting hand; with the result being a stiff plodding playing style.