Hi! 
Good question, and obviously a very relevant one! As many of us will have heard, one of the simplest summaries of the Buddha’s teaching and one that has been quite popular in Buddhist history is that the Buddha taught phenomena arisen from causes and their cessation.
Let’s take a look at the Buddha’s statement again.
“Mendicants, the conditioned has these three characteristics. …
“Tīṇimāni, bhikkhave, saṅkhatassa saṅkhatalakkhaṇāni. …
Arising is evident, vanishing is evident, and change while persisting is evident.”
Uppādo paññāyati, vayo paññāyati, ṭhitassa aññathattaṁ paññāyati.”
One way of taking this, which I would suggest is not what the Buddha intended, is that the Buddha is just looking around him and sees things arising and ceasing then makes a broad generalization to everything.
I’ll call this the ‘inference by generalization,’ and as I said, I don’t think this is what the Buddha is doing here. However, if we relate to the statement as a broad inference by generalization, it’s only natural to ask: but is that fool-proof? Is that a safe generalization to assume? Obviously, many spiritual and philosophical traditions would argue that there are eternal things without an arising, changing, or ceasing, based off their own inferences and experiences. So is it a matter of personal intuition? I suggest no.
What alternatives might there be then?
There are two main approaches to this. One is logical necessity, and the other is meditative realization. These two may seem separate, and although they are not equivalent, they are very closely connected. Let’s start with analysing the first of these two.
The absolutely key word here, which is given especial emphasis in the Pāli-language original, is saṅkhata. The Buddha is not offering a truism such as “everything changes.” This discourse is much more specific, and it is saying that there are three characteristics that mark whatever is saṅkhata .
Now, in the most general terms as apply here, ‘saṅkhata’ means ‘produced by causes.’ So the Buddha is saying: “Whatever is produced by causes is characterized by (1) an arising, (2) change while persisting, and (3) cessation.” Then to your question: on what grounds?
If something is produced by causes, it means that it comes into being only when all necessary causes are present. We can sum this up with the singular “cause” meaning whatever will altogether produce a given “effect.” In fact, we could even translate saṅkhata as ‘effect’ in this sense.
So then, all effects by definition have causes, and all effects by definition only exist when their causes exist. This is simply a given in the term itself, to which the three characteristics are added as the Buddha’s contribution.
Whatever, not having been, comes into being only when the cause comes into being, by definition has an “arising.” Any effect, by its very nature, arises. It is not an absolute foundational reality, but something which arises under certain conditions.
If something can come into being, then it means the cause producing the effect is also not eternal. If the cause were fundamental and absolute, the effect would not come into being. But because effects come into being only when the causes are present, the causes themselves have an arising.
And this is where one of the Buddha’s big punchlines comes in. If effects only arise from causes, and causes themselves are not permanent, then those causes can also cease, and therefore the effects cannot be permanent. Whatever arose did not exist before it arose, meaning only under certain circumstances does it come into being. Because those circumstances themselves are temporary, they can also change, and so whatever arises must also have the nature of ceasing. Just by its very nature, included within the implications of something being caused.
X depends on Y circumstances to arise. Y circumstances are not ever-present, but can come into being dependent on other circumstances. So when X comes into being dependent on Y, it can also go out of being when Y goes out of being as it was before.
The last aspect of this is the middle one, change while persisting. This we could say is less directly evident, but could be included under the fact that the effect must continuously be kept in being dependent on causes, and because the causes themselves are conditional, then even during the lifetime of a given effect the causes have to be continually renewed, and so they can also change within certain parameters.
In other words, something that has a beginning also has an end, but it also has a middle where it must be bound by time even during its lifespan. Because it’s bound by time, it’s in a state of change. This can vary obviously. Something like water depends on temperature, but the temperature can change within a certain scale until eventually the causes are so drastically different that it becomes ice or gas and we say the water has ceased.
So this is the main point of the Buddha’s statement. If something is an effect, if it has a cause, then it cannot be eternal. It comes into being, so by that very fact it can go out of being, and it will also be bound by time and change during its lifespan. The Buddha is putting parameters around what is eternal or not, and he’s making it clear that effects cannot be eternal. This is confirmed by the teaching immediately following this one at AN 3.48:
“The unconditioned has these three characteristic s.
“No arising is evident, no vanishing is evident, and no change while persisting is eviden t.”
The Buddha is contrasting what is an effect of causes from what is not an effect.
This brings us to the second part, which is asking how one can have a meditative realization of that fact. And, in the early texts, this is where the Buddha hones in on what “effects” in particular he is emphasizing.
In discourses like SN 12.20, the Buddha says that what is saṅkhata are the standard twelve factors of dependent origination, such as rebirth, consciousness, the senses, existence, etc. Another example would be the list of the five aggregates or the six senses. In other words, the Buddha is now focusing on the effects relevant to suffering: existence, rebirth, the mind and body.
If you apply the same principles as above to the round of rebirth and existence, then you ask: do states of existence have a cause? Do they come into being? Does the body have a cause? Does consciousness have a cause? If they have causes, they are effects; they are dependently arisen; they are produced and made. And then you return to the Buddha’s statement above that whatever is produced by causes comes into being and by that very same nature it will be bound to change and to going out of being.
So this is where the idea of an eternal heaven or an eternal mind or an eternal God or an absolute reality come into question. If we are not in heaven now, and then because of a change in circumstances a heavenly existence arises, well then that heavenly existence is an effect, it has a beginning, it is dependent on causes which are temporary. You investigate the root.
So if you develop meditation, then you can look into the mind and ask if the mind has a cause for its continuation. And if so, what would happen to the effect if the cause were no longer there?
In other words, it’s a matter of figuring out what the limits of the ‘produced (sankhata)’ is. If something is produced, if it depends on causes, well then it won’t be a resting place according to the Buddha. Many people would agree, but they might disagree on the limits. They might say bodies are dependently arisen, but the mind is not. So you investigate: does the mind depend on causes? The same is true of anything. The Buddha delimits what phenomena are arisen from causes, and he shows how those phenomena therefore are subject to cease when the cause ceases.
With that, you can start making some sense of the famous statement referenced above that led to Ven. Sāriputta understanding the Dhamma:
“Whatever phenomena arise from causes, the Tathāgata has declared their cause as well as the cessation of these: such is the doctrine of the great ascetic.”
That is how someone can personally realize, not by vague generalization but by sheer necessity, that the cycle of rebirth and existence can cease. In other words, that samsāra is impermanent. And of course that has various implications and consequences. It is the Buddha’s core discovery, what he described in his teachings on dependent origination and Nibbāna, and what he thought was most profound after he attained awakening:
[The Buddha thought:] ‘This principle I have discovered is deep, hard to see, hard to understand, peaceful, sublime, beyond the scope of logic, subtle, comprehensible to the astute. But people like clinging, they love it and enjoy it.
‘It’s hard for them to see this topic; that is, specific conditionality, dependent origination. It’s also hard for them to see this topic; that is, the stilling of all activities, the letting go of all attachments, the ending of craving, fading away, cessation, extinguishment.’ (MN 26).
Hope that is more clarifying than confusing! 
