The concept of being the part of the universe

I don't think I'd put it like that ("being one with the universe") but here's my guess as to what that may mean:

• Buddhism is more about mind than about matter. It's not about Physics (atoms and stars and so on).
• Buddhism nevertheless claims to describe various laws of nature (see e.g. here), and by understanding these 'laws' (which describe how the world is) we can better attain peace, even lasting peace, liberation.
• A lot of Buddhism is to do with ethics (see also e.g. here or here), especially to begin with -- of which an example are the five precepts (and I think of the the brahmaviharas too as "ethical") -- hence (by being ethical or virtuous) you might hope for social harmony, being "one" with society
• The Buddhist "four noble truths" teach that people suffer when they crave for things to be other than as they are -- I first learned that as "craving to have what they don't have, craving to keep what they can't keep" -- the act of not craving for things to be other than as they are, of avoiding a split between "how it is" versus "how I want it to be", is a type of unity. Whereas the opposite (i.e., craving) is a kind of split between the universe and the self, e.g. "the universe says X but 'I' want Y instead" creates a separate 'I'.
• It's not all apathy, though -- there is such a thing as good and bad, right and wrong, so there is such a thing as "right effort" for example. Part of the way in which you don't crave for things to be other than as they are is to behave well, do good -- and to "purify the mind", maybe to unlearn any bad habits -- so that you have no cause for regret, no remorse.
• There's some moderation, or "a middle way between extremes", in Buddhism -- e.g. though monks and nuns may be penniless, laypeople hopefully support them with food and clothes and medicine, the necessities.
• Some Buddhists emphasise that inter-dependence further -- see e.g. interbeing.
• There's a lot of sensible advice -- for example, "have good friends, good teachers, not bad friends".
• There are various forms of meditation to calm the mind; and/or to train the mind (to practice better mental activity, for example kindness rather than any anger); and/or to avoid getting upset while the mind is active; and/or to see the world "as it truly is" (perhaps without too much additional mental baggage).
• There is a "non-self" doctrine or principle (or characteristic) in Buddhism, which maybe long to explain, but perhaps that (i.e. training to be sort of less selfish) is part of being "one".

In Western terms, the usual context for this could be explained as determinism, accepted in its positive sense.

Our default model of life entails this notion of "beings" or "organisms" freely moving in an otherwise deterministic or at least probabilistic universe. These beings are supposed to have a unique ability to act at their own will, as opposed to being moved by the causes and conditions as the rest of the universe.

From Buddhist perspective, this special status of separation is understood to lead to both ethical as well as psychological problems. One simple solution is to abandon the default model and adopt a different paradigm in which our actions are understood to be continuous with the causes and conditions in the objective universe. This is what's generally known in the Western philosophy as determinism.

However, from Buddhist perspective, this perspective is not equivalent with fatalism, which is in fact sharply criticized by the Buddha. Instead, determinism is understood to be the "external" perspective on things, while from the subjective perspective each individual sentient being still has the job of making the best possible choice out of the available possibilities. Indeed, this is the same exact situation as is faced by an intelligent AI-driven robotic agent (or a virtual agent such as an NPC), which has to make decisions about its actions despite being a part of the otherwise fully deterministic environment (the computer system).

So, in Buddhist practice, there is this progression from the initial assuming that one has free-will but in fact being constrained by all kinds of fears and insecurities limiting one's spontaneity and will-power -- to fully realizing that one is an integral part of the fully deterministic system, while at the same time opening up to spontaneity and subjective freedom by breaking through fears and insecurities, and by developing the will-power to counteract the pathological & destructive types of external influences. The resulting state is one where will-power, control over one's mind and mood, a sense of no fear, are co-joined with the fully clear understanding of the deterministic rules governing the universe, resulting in a state of wise, enlightened, optimistic spontaneity of one no longer in conflict with the causes and conditions of the objective universe.

All of the above is not the "High Buddhism", but rather one practical aspect of it that has to be mastered. What I call "High Buddhism", is understanding (and practical realization thereof) that such concepts as "organism", "universe", "all" etc. - are mere constructs that are defined arbitrarily and relatively to whatever axioms and contexts we happen to choose. The very act of defining these concepts places certain restrictions on us, making us bound to operate within the definition of the world framed in terms of such concepts. The "High Buddhism" is clear understanding of how framing the experience affects our sense of what's going on and the possible acts. When this understanding is reached, one realizes in practice that spontaneity vs determinism is in fact an artificial dichotomy that only exists because of the conflict between two theoretical descriptions of how things work. To become actually free from all such descriptions is therefore the necessary step to Final Liberation.

At the end of the day, the state sought by Buddhist practitioners is the unity of harmony with the (deterministic) forces operating the universe -- on one hand -- with subjective freedom to choose our own interpretation of what's going on -- on the other. It's a state of indescribable freedom which is not in any (subjective nor objective) conflict with the way things are.

This is what's understood under "being the part of the universe", according to my studies and experience gained from practice.

"I mean are we born from universe and when we say the word "universe", what exactly do we mean?

The idea (or what is discovered) is that we are in a sense born from the universe (in a way this is obvious) but it may be better to say we are the universe. It could even be said we are not born at all. Even the statement 'we are' would be denied.

If 'Universe' means 'all that exists' then this is not a Buddhist meaning of the word since it would leave out the source of existence. This source would real where all else would be created thus not metaphysically or independently real. The Buddhist 'universe' is bigger than the universe of physics since it includes the totality of Reality and Appearance or both of Nagarjuna's 'two worlds', but also smaller since space-time and its contents would be reducible.

Hence the Buddhist Universe is a fundamental concept where the scientific concept is non-reductive. This would be why Buddhism has a coherent metaphysic while physics cannot find one. How we define 'universe' is therefore important. If we follow the scientists and say it is 'all that exists' we will have no hope of a fundamental world-theory.

You may in fact be asking about "identity view", which is a key concept in the suttas:

For a little baby doesn’t even have a concept of ‘identity’, so how could identity view possibly arise in them? --MN64

With identity view (i.e., "I am") comes a host of problems. Indeed, you might say that we get a "universe of suffering". MN64 discusses identity view at length. For example, with the notion of self and a corresponding need to perpetuate self (e.g., "I like chocolate") comes suffering (e.g., "GIVE ME CHOCOLATE NOW!").

Without identity view there is no longer a gap between a self and the universe. The self is found to be the gap, the hiccup, the breach in existence. Without identity view there is equanimity, which is often what folks generally mean when they say "one with the universe."

‘This is peaceful; this is sublime—that is, the stilling of all activities, the letting go of all attachments, the ending of craving, cessation, extinguishment.

There still is chocolate. But the happiness of eating chocolate is no longer a mandate of craving. If you eat the last chocolate, fine. The last chocolate is eaten.