How can I believe that I exist?

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René Descartes grappled with this question way back in the 1600s. His main contribution to philosophy was in shifting the debate from “what is true?” to “of what can I be certain?” The way he did this was to come up with a methodology for inquiring into the unknown. These methodologies are now known as Cartesian doubt.

First you doubt everything, then try to construct something incontrovertibly true from basic principles. The idea was so new at the time that he’s widely considered to be the father of modern philosophy.

I’ll summarize the basics of his second meditation here:

We have access to only the world of our ideas; things in the world are accessed only indirectly.
These ideas are understood to include all of the contents of the mind, including perceptions, images, memories, concepts, beliefs, intentions, decisions, etc.
Ideas and the things they represent are separate from each other.
These represented things are many times "external" to the mind.
It is possible for these ideas to constitute either accurate or false representations.

From Wikipedia.

It is this meditation that spawned the famous “I think, therefore I am.” This position is called dualism and formed the basis for the mind-body problem that raged over the next few hundred years and still has unanswered aspects to this day, the most notable of which is the ‘hard problem of consciousness’.

His third meditation concerned God, he concluded that he could only exist because of God. His reasoning may be interesting to you, if unconvincing to the positivist atheist. It’s a kind of window into the minds of thinking men of the day.

The reasoning goes like this: There are three types of ideas, those that come from within us, those that are invented by us, and those that come from experience in the world. He reasons that God must be one of the ones that comes from within us. God is perfection, the idea of perfection could not have come from anything that was less perfect than God, therefore Descartes comes from God.