The Sun and Moon are part of the primary triad in astrology (along with the Ascendant). The triad are the most important parts of your natal chart. They are the most closely tied to your personality, to who you are as a person, and your most basic self. So it should come as no surprise that they're also very important in love astrology!
What do the Sun and Moon rule in astrology?
The Sun is your inner self, the you who comes out when you're around those who know you best and you're comfortable. The Ascendant is your external self, the you that others view you as that you don't know too well. This is why most of us come across differently from how we really are. Because the Sun is who we really are!
The Moon is your emotional self. It rules how you express yourself emotionally, and what you need for emotional stability and security. The Moon shows how you nurture and emotionally connect to the world, and your core self. So, you can see why these two are very powerful.
Why are the Sun and Moon important in relationships?
We often connect with people in our lives who connect with our Sun or Moon; or conversely, we connect to their Sun or Moon. Connections to the Sun or Moon show a connection to who we really are (Sun), or show an emotional connection (Moon).
This is usually strongest when you have a Sun-Moon connection, meaning one person's Sun is in the same sign as the other person's Moon. The strongest is when they conjunct, or are in about the same spot degree-wise. You might find that people whose birthday puts their Sun around the location of your Moon are people you continually gravitate toward.
For example, if your Moon is at 17 degrees Aries, you may gravitate toward people born between April 5th and 9th, which places their Sun with your Moon. The Sun is in about the same location in the Zodiac at about the same time each year, so you can look up in an ephemeris which date conjuncts your Moon location. Don't be surprised if you find that you've connected with many people with birthdays around that date!
This is very helpful in relationships. It gives an understanding between both people in the relationship. It's not automatically romantic, and can show helpful energy in relationships with your work, in your family, and with your friends. It's not in-your-face because there isn't any friction (if there is friction, it's the result of something else). It's easy and unconscious.
You can also see this with house placements. So say your natal Sun is located in the 9th house and their natal Moon is located in the 9th house - you'll experience the same kind of connection. You can also see it with the sign the house naturally connects to - so if your natal Sun is in the 9th house, that naturally connects to Sagittarius, so you may link up well with a Sagittarius Moon.
And, you can see it with planetary conjunctions - so say your natal Sun is conjunct Saturn, this would help you link up with someone who is a Capricorn Moon, which is naturally ruled by Saturn. Lots of options!
Suggested Reading: When Venus Meets Mars