ADD: I see there is more than one part to this question.
First, for the mathematical aspect:
The term "laws of nature" or "physical laws" refer to what a mathematician would call the evolution map of a dynamical system.
A "dynamical system", or DS, is basically a combination of a space $M$, typically some form of manifold, which is called the phase space and for which elements of represent possible present states or configurations of the dynamical system in question, a space $T$, called time space, which in this case is typically the real numbers $\mathbb{R}$ but in some cases is taken as a discrete space such as the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, and represents the possible values that "time" can have, and finally a map $\Phi$, which is called the dynamics or "physical laws", if you want, of the system (also, time evolution map). The map $\Phi$ is just a mathematical function (though we might pose "reasonableness" stipulations like continuity, differentiability, etc.), and one which accepts both a time increment $\Delta t$ from $T$ and a configuration point $p$ from $M$, and returns what that point will evolve into after the lapse of $\Delta t$ worth of time. We typically write such as
$$\Phi^t(p)$$
or
$$\Phi(\Delta t, p)$$
depending. The key point is that also, this map can't just be arbitrary - in particular, it must satisfy two rules:
$$\Phi^0(p) = p$$
i.e. evolution by no time at all leaves the point unchanged, and
$$\Phi^{\Delta s}(\Phi^{\Delta t}(p)) = \Phi^{\Delta t + \Delta s}(p)$$
which basically says that if we evolve first by a time amount $\Delta t$, then evolve it some more by another amount $\Delta s$, that the total evolution produced should agree with that by evolving it straight off the bat by the sum-total of those amounts: $\Delta s + \Delta t$. The exponent notation is the one I personally prefer because the latter identity shows that this can be thought of as a sort of generalized compositional power or iteration of a function, so that the iteration count, or number of repeated applications of the function, may be a continuous parameter.
To formulate the concept of time symmetry in classical mechanics here requires a bit of care. For one, one should easily see from the above that an evolution map $\Phi$ as just defined will always respect a time translation, if you can also have negative times as well as positive. Thus, we need to be a bit choosey in what we call the manifold $M$. Generally, we should choose $M$ to also contain a time coordinate in itself - this can be understood by keeping in mind that the "time" in $\Phi$ actually is not a coordinate time, but rather a time increment given for evolution. If we want a time coordinate system, then we have to supply that in $M$. For example, for a single-particle classical system, we may take $M$ to be the set of all points
$$(t, \mathbf{r}, \mathbf{p})$$
where $\mathbf{r}$ and $\mathbf{p}$ are position and momentum and $t$ is the time. We may, say, define the dynamics of a free particle (i.e. no forces) thusly, where we take $p$ as having the form above, and $m$ is a pre-specified mass constant:
$$\Phi^{\Delta t}(p) := \left(t + \Delta t, \mathbf{r} + \frac{\mathbf{p}}{m} \Delta t, \mathbf{p}\right)$$
You should check that this satisfies the above laws to be a dynamical system. Basically, what it says is that a forward advance by $\Delta t$ time units should increase the time coordinate by $\Delta t$ (as should be expected), and it will move the particle by the amount given by its velocity maintained steadily over the given time.
Time symmetry now means this: If we take the time translation map
$$T_{\Delta t}(p) := (t + \Delta t, \mathbf{r}, \mathbf{p})$$
which shifts the spatial part of a given state $p$ up/down through time, then the evolution of that state, forward and backward, should be a time translation by the same map as the original state - that is, the evolution does not depend on the time coordinate assigned to the state, only the state itself:
$$\Phi^{\Delta s}(T_{\Delta t}(p)) = T_{\Delta t}(\Phi^{\Delta s}(p))$$
for every evolutionary increment, forward and backward, $\Delta s \in T$. Mathematically, the time translation and dynamical maps must commute. This is clearly a non-trivial statement since while that our evolution map ("law of physics") always must give the same evolution for any $M$-point and increment, a "point" now includes a time coordinate. For example, we may imagine that at a set time coordinate, instead we have that the particle in question receives a blow. The system is clearly not symmetric with regard to the time translation of the ball's evolution alone (i.e. ignoring the agent giving the external blow as being part of the system) and, of course, its energy is not conserved (which is accounted for in the real world by noting that an external blow makes it an "open system" and thus receives energy from that actor).
Second, for the philosophical aspect:
There is no reason we should "assume" it. We could, after all, live in a Universe where that, for example, on Mondays things decide to fall up for 1 second at the beginning of the day, then they drop back down again. That'd be very bad for us as we are now (though if this were our world, we'd have to be evolved/built in some fashion to take the "punishment", so then would not really be as "bad"), but it is an entirely logically consistent way that things could be. In this case, the dynamical map describing the physics would not translate as just mentioned - those "fall-ups" would act like the "blow" I talked about before, but they would not be the result of an extrinsic agent and rather intrinsic to the laws on which the Universe operated. If you shifted such an evolutionary path up/down on the time axis so that the "fall-ups" no longer coincided with the beginning of Monday, you'd have a situation that violated those laws because the laws say they must happen at that particular time.
The pure fact of the matter is we have no other reason, at least scientifically, to assume this other than that, from what we've observed, our Universe looks to be constructed to have no "preferred" points in time of this sort. We don't, at least as of yet, have any uncontested access to the mind of its Creator(s), or even can make an uncontestable case that any such do or do not exist, and/or what non-mental alternatives might be responsible for its existence in lieu of them, thus we have no way to answer this.
Finally, for the "differential equation" aspect:
This is really just a corollary of the mathematical definition. If the evolution map is given by a differential equation in the time coordinate, then for the map to yield the same trajectory after an arbitrary temporal translation, it will, logically have to be given by the same differential equation at all points in time.
