There is a short answer involving Newtonian ideas and flat space, and a longer answer involving General Relativity.
On a historical note, let's also recall that both Lemaitre and Hubble deserve mention when thinking about the early work here; Lemaitre first clearly enunciated the idea we now call Hubble's law, and estimated a value for the constant we now call Hubble's constant. Hubble contributed greatly to the experimental methods and observations and making the idea take hold.
Now let's present the Newtonian argument. Suppose an observer at one place finds that galaxies are moving away from him with a velocity whose size is some sort of function of distance away, but we don't say what function yet. So we have
$$
{\bf v} = v(r) \hat{\bf r} = \frac{v(r)}{r} {\bf r}
$$
where $\bf r$ is the vector from the observer to the galaxy.
In particular, for two galaxies A and B we get
$$
{\bf v}_A = \frac{v(r_A)}{r_A} {\bf r}_A,
\;\;\;\;\;\;
{\bf v}_B = \frac{v(r_B)}{r_B} {\bf r}_B
$$
It follows that the relative velocity of galaxy B with respect to galaxy A is
$$
{\bf v}_B = \frac{v(r_B)}{r_B} {\bf r}_B
- \frac{v(r_A)}{r_A} {\bf r}_A,
$$
For most functions $v(r)$ this has no simple expression. But suppose the function is
$$
v(r) = k r
$$
for some constant $k$. In that case we find
$$
{\bf v}_B = k {\bf r}_B - k{\bf r}_A = k ({\bf r}_B - {\bf r}_A) = k {\bf r}_{AB}
$$
and this applies for all galaxies B. The vector ${\bf r}_{AB}$ here is the vector from A to B.
Therefore we have found that an observer on galaxy A finds that he too observes the same effect: other galaxies are moving directly away from him, and with a speed proportional to their distance away.
In this way observations from a single location can be used to infer what would be observed from other locations, and in the case of a linear relationship between $v$ and $r$ the whole situation has this uniformity with respect to position. Other relationships would not be so simple. In the case of a bomb exploding, for example, if some material moves away quickly and some more slowly, then typically the front material begins to slow as it encounters air resistance and the relationship between speed and distance away at any one time is not any simple function. The Hubble expansion is not like a bomb exploding, and indeed, as we have seen by the above derivation, there is no unique centre. All positions can equally well serve as the centre, which is the same as saying there is not really any centre. Many other examples of this sort of expansion can be found in thermal physics: think of a loaf of bread expanding in the oven, for example, or more generally practically any uniform solid undergoing a thermal expansion.
That is the "short" answer.
The longer answer is that similar ideas apply in a full treatment employing General Relativity. The main difference is that you can no longer obtain relative velocities just by subtracting one galaxy's velocity from another. Instead the typical way to proceed is to propose a model in which there is a global expansion of a homogeneous fluid, and then show that this is consistent with observations. The "fluid" here is the whole universe; "particles" in the fluid are galaxies or galaxy clusters.
Finally, note that the proportionality constant $k$ is a constant in the sense that it is the same for the whole universe at any given time, but it can in principle change with time, and the expansion would still have the same uniform quality, but now it would proceed more rapidly at some times than at others. This is why the proportionality factor is called the Hubble parameter rather than the Hubble constant, when we want to consider the long-term evolution of the universe.
