Share button.
Before trying this, if you don't use the recording feature for anything else, make things easier on yourself by setting the recording limit to something small. To do this, press the share button to go into the display menu and go to Sharing and Broadcast settings > Video Clip settings. Set it to 30 seconds or 1 minute. Then just play. This will keep you from having to trim down the video from being too long, and just adding another unnecessary step.
If you see something that you didn't like, press and release the share button to bring up the display menu and press Square to save the previous gameplay (either 30 seconds or a minute if you did what I suggested above). This video will show up in the MLB 17 section of your capture gallery.
From here you have a couple of options, though they both ultimately lead to the same result. You can either upload to Youtube directly from the PS4 or save the video to a flash drive. For simplicity's sake I'd suggest uploading straight to Youtube. Find the video in your capture gallery, highlight it and press Share. Then just follow the instructions on-screen (it will walk you through creating a Youtube account if you don't already have one). When the process is done you'll be able to see the video on Youtube's site, and the URL where you can watch that video is your video evidence. (you can also embed so the player actually shows up in the post, but that's getting into unnecessary stuff again)
I would suggest not using the in-game replay for this - live recording of the actual play is always best.