I always wonder how all this started... How did we arrive here? What are we doing in this place?
So are there teachings in Buddhism that are clear on this point?
How did the world start? and How is it gonna end?
Here is one teaching the Buddha gave on the start of the world:
Dwelling at Savatthi. There the Blessed One addressed the monks: "I will teach you the origination of the world & the ending of the world. Listen & pay close attention. I will speak."
"As you say, lord," the monks responded to the Blessed One.
The Blessed One said: "And what is the origination of the world? Dependent on the eye & forms there arises eye-consciousness. The meeting of the three is contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging/sustenance. From clinging/sustenance as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. This is the origination of the world.
And in the same teaching the Buddha gave the explanation for the end of the world:
"And what is the ending of the world? Dependent on the eye & forms there arises eye-consciousness. The meeting of the three is contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. Now, from the remainderless cessation & fading away of that very craving comes the cessation of clinging/sustenance. From the cessation of clinging/sustenance comes the cessation of becoming. From the cessation of becoming comes the cessation of birth. From the cessation of birth, then aging & death, sorrow, lamentation, pain, distress, & despair all cease. Such is the cessation of this entire mass of stress & suffering. This is the ending of the world.
To understand this I think takes familiarization with the Dharma and specifically the teachings on the Four Noble Truths and dependent origination.
how all these started?
There's no real starting point. But if you are asking the origin of earth and how it's populated, how we came here etc. you can read the Agganna sutta. We basically came from a Brahma realm called Abhassara. We were luminous beings. Not at all like monkeys. Only after a long time we changed into this monkey like form because of our behavior and what we consumed.
What are we doing in this place?
That's up to you to decide.
How it is gonna end?
Read the Satta Suriyuggamana sutta or Satta Suriya Patubava sutta. I don't have a link to a good English translation. Basically, it will end in fire this time. According to the Sutta, 7 suns will appear and burn the earth to a crisp. Some translate it as the sun expanding 7 times. But, given that the Andromeda galaxy is heading towards the Milky Way for a big clash before our sun dies, it's not unlikely that 7 suns will appear after about 4 billions years and burn us to oblivion.
To add to Yeshe Tenley's answer, the origin of the world is explained in Pali Canon's teaching on Dependent Co-arising, also known as the Twelve Nidanas. It all starts with Ignorance and proceeds to Craving (and Aversion) which leads to separation into "self" and "the world".
While Pali suttas don't go in too much details, this process of separation is further explained in the Tibetan tradition. Here is, for example, an abridged quote from Lamp of Mahamudra by Tsele Natsok Rangdrol:
Your natural essence [...] is neither tainted nor spoiled by such designations as pleasant or unpleasant, being or not being, existent or nonexistent, permanent or annihilated, self or other, and so forth. Because it is not established as a certain kind of identity, your essence can serve as the basis for the manifestation of any form or conception to manifest. Yet, no matter how it manifests, ultimately this essence has no true existence. Thus, it is a great emptiness free from the limitations of arising, dwelling, and ceasing.
...Due to the ignorant aspect of this neutral all-ground, you do not cognize your own essence, and the natural state is not realized. In this way have you obscured yourself. Called "coemergent ignorance" or the "great darkness of beginningless time" since it is the basis from which all disturbing emotions and deluded thoughts arise, it is also known as the "all-ground of various tendencies." Hence it is the ground of confusion of all sentient beings.
Craving and separation
...From this coemergent ignorance arises a fixation on an ego and self-entity. On the basis of this "self" arises the fixation on "other." Not recognizing this personal manifestation for what is, a personal manifestation, one grasps at it as though it were an external object. [...] It is the cognitive act of confusing object and mind as being separate and is accompanied by the forty thought states resulting from desire, such as attachment and clinging.
The origin of the world
...Through the power of being habituated to confused fixation, your personal manifestations will appear as a world with inhabitants. [...] The various objects of gross materiality and the pure and impure parts of nadi, prana, and bindu of the inner body, as well as all the various phenomena of samsara and nirvana, the worlds and beings of the three realms, appear externally in an interdependent manner.
I take these statements to be true:
In other words, these are among The unanswered questions.
I prefer this to assuming that every conceivable question has a good answer -- see also for example Simsapa Sutta (SN 56.31).