This is actually something I've been really pondering for a few months now. I've been getting away Christianity, the religion I was raised in (this makes me sad, but I can't arbitrarily choose to believe in something, even if others seem to be able to).

There is something that religious people feel as God. I've felt it, and I didn't imagine it. The feeling, whatever it is, is real. I've recently gotten into more Eastern type stuff, and I felt that feeling again. But this time, the book I was reading described it as my higher self, or as simply "being". And this time I could realize that there was an element of me in the feeling, but it felt somehow greater and more permanent than me at the same time. To reach that place, find a way to quiet your thoughts, images, emotions, and general mental noise to a point where you can see what lies underneath it, and you'll see that there is a stillness there, a peaceful awareness, that is you, that is like the sky into which the clouds of your thoughts are placed. If you can connect with this part of you, you'll notice love, joy, and peace emanating from it. The importance of your day-to-day worries will start to diminish. And you'll feel connected with something more permanent, more eternal, than the "I" that your mind builds out of the narrative of thoughts and feelings you generate.

There is a concept in some Eastern circles called "non-dualism", which is basically the concept that the "self" is not a separate entity from the rest of the universe. That the universe is really one entity, and "our" personal boundaries are completely arbitrary. This leads to really strange quotes from non-dualist thinkers:

In other words, the so-called involuntary circulation of your blood is one continuous process with the stars shining. If you find out it’s you who circulates your blood, you will at the same moment find out that you are shining the sun. Because your physical organism is one continuous process with everything else that’s going on. Just as the waves are continuous with the ocean. Your body is continuous with the total energy system of the cosmos, and it’s all you.

So that's another way of looking at connecting with your "higher self".