Why does God exist as opposed to nothing existing. Is this just a contingent fact that we must accept or is there some kind of reason we can understand a bit why it must be so? Please source.
According to the OP's parameters:
A) G-D exists.
B) Nothing exists.
Since we are having this conversation, B is obviously false.
Therefore A is true.
If we define G-D like the Rambam (and others) defines Him, then there is no question at all. G-D is the one true entity. If everything was to cease besides Him, there would only be Him. He is not affected. If He were not to exist, then nothing would exist, because everything's existence depends on Him. He is eternal and was always there. (see Rambam, Yad HaChazakah, ch.1)
"Hashem, He is G-d, there is nothing else beside Him." Devarim 4:35
Therefore, once it is true that there is only G-d and He is eternal etc.; one cannot ask why it is that way. It simply is. If the alternative of "nothing" were the reality, then it could also not be asked why it is so. It would just be nothing. There are no chances of either. It either is A or B.
So the OP is correct: "Is this just a contingent fact that we must accept ..."
There is no why.
The first three laws in the Rambam's Code (Yesodei haTorah 1:1-3):
יסוד היסודות ועמוד החכמות לידע שיש שם מצוי ראשון והוא ממציא כל נמצא וכל הנמצאים משמים וארץ ומה שביניהם לא נמצאו אלא מאמתת המצאו.
The foundation of foundations and the pillar of wisdoms [written in four words, initials "Y-HV-H"] is to know that there is a First Existence, and He brought into existence all that exists. Everything that exists from the heaven and the earth and what is between them do not exist except from the Truth of His Existence.
ואם יעלה על הדעת שהוא אינו מצוי אין דבר אחר יכול להמצאות.
If it would cross your mind that He does not exist, nothing else would be able to exist.
ואם יעלה על הדעת שאין כל הנמצאים מלבדו מצויים הוא לבדו יהיה מצוי ולא יבטל הוא לבטולם שכל הנמצאים צריכין לו והוא ברוך הוא אינו צריך להם ולא לאחד מהם לפיכך אין אמתתו כאמתת אחד מהם.
And if it would cross you mind that every existing thing other than Him would not exist, He alone would exist, and He Himself would not be annulled ever. For everything that exists require Him, and He (blessed be He) does not require them -- not one of them. Therefore His Truth is not like their truth.
We exist contingently. We happen to exist. But Hashem's Truth is not like our truth, Hashem necessarily exists.
To ask "why" means that there were other possibilities and we want to know what caused one and not the other. But Hashem has no cause, and no other possibilities but Existing.
(By the way, the Rambam has a proof of Hashem's Uniqueness and Indivisibility based on the idea that anything that has parts only exists contingent on those parts being aligned to make the whole. But we defined G-d as the initial necessary Existence which causes all the derived existences.)
I'm paraphrasing the Rambam's teachings in the propositions and first couple chapters of the second book of the Moreh. Essentially, that which passes from potentiality to actuality need something that already exists in actuality to effectuate that change. That which doesn't pass from potentiality, but always exists wholly in actuality, does not have any cause whatsoever. In plain English, things that change must be caused to change by something else, and that which doesn't ever change cannot be said to have any cause. A cause here is defined as an answer to a "why" question.
Aristotelians such as the rambam recognize four different types of causes. Material causes, formal causes, efficient causes, and final causes. Only efficient causes and final causes are pertinent to your question, so I'll ignore the other two. If I was to ask why is there a table, I could give two types of answers. A carpenter built it (the efficient cause, what did it). Or I could say, to hold food (the final cause, what its purpose is).
Your question, why does G-d exist as opposed to nothing can be rephrased as two different questions. 1) What caused G-d to exist, or 2) What is the purpose of G-d existing. The first is looking for an efficient cause, the second is looking for a final cause. However, G-d does not change. He is pure actuality with no potentiality. Therefore, you can't answer these questions. We know G-d is a pure actuality because everything which changes must have been changed by something else. For various reasons which aren't applicable to your question, this terminates at something which itself doesn't change. This is G-d, who has no causes as a matter of definition. Therefore your question doesn't make sense as it can be rephrased as "what is the cause of the thing that has no causes?"
This is called a categorical error. You can't answer these questions, not because they are a mystery, or only G-d knows, or any other unsatisfactory answer. The question asks for an answer that is inapplicable to the subject. Another example would be like asking how much the color blue weighs. Color is not subject to weight, so the question doesn't make sense. G-d is not subject to causation, so the question doesn't make sense.
Whether God exists or not is not valid question. Einstein talked in God terms but he probably didn't mean what any Rabbi means when he speaks about it. The question should be - "Has God spoken his word" - or has he come in intelligible interaction with us.The answer is Yes because our ancestors told us so, just as in case with any other historical information passed through generations.