H4 should be plug and play. The connector you've pictured looks to be a harness adapter of some kind. Take
@C.J. Roberts recommendation. Follow them back. You should find the OE H4 plug that will mate up to your new lights. If not, someone replaced the OE harness.
Switching to H4's is certianly better that the OE 'candle in a sealed beam' but the LEDs win IMHO. I will tell you with 100% confidence, you get what you pay for. Inexpensive Chinesium LEDs are very often a scattered mess (always in my experience with three different bargain LEDs). I had a set of REALLY cheap LEDs installed by the OP basically as "buy-bait". Replaced them with a frequently recommended LED housing, hated them and then sold those in favor of H4's, which I lived with for a while. Replaced those with a quality set (IMHO) of LEDs that I am thoroughly happy with them.