You can see it like this: if you consider the universe as a whole, "of course" each process is "spontaneous" in the sense that the total entropy of the universe entropy always increases. After all, if something happens somewhere in your universe, then it must be allowed to happen: otherwise, it just would not happen!
However, what you ususally care about is that your fridge stays cold, and that is why thermodynamics focuses on sub-systems: it is not enough to say "somewhere in the universe there is the possibility of a fridge". You want to know about your fridge (and your electrical bill at end of the month and the natural resources of your planet!).
A question about a spontaneous process would be: if I buy a fridge, would it get cold? And the answer is: no (unless you plug it to an external power source)! If you turn the question to "do cold fridges exist in the universe without any external assumption except the existence of the universe" then the answer is, of course, yes.
So depending on the scale you look at things (fridge < electrical bill < planet < universe) things can be considered spontaneous or not. You need the current for your fridge, you need a power plant for the current, you need the sun for the power plant, you need the big bang for the sun, etc.). So you need to choose: at what scale do I look at the system: would a fridge work without current? Would current work without the sun? Would the sun work without the univse?
Can you say this in more scientific words?
The universe as a whole only has processes which increase the total entropy. Now, let us focus on the entropy of a sub-system!
Here is the tricky point: a sub-system of the universe, if you consider it as an individual object, could locally decrease its entropy if you supply work to it, i.e. it could to something you would NOT expect to happen unless you considered it as part of a bigger system.
So, let's say I have a subsystem, I see its evolution. Then the question is:
do I have to resort to the presence of an outside energy source in my system to explain its behavior?
If yes, then the process is not spontaneous. If not, then the process is.
A spontaneous process inside a system the you are observing is one that you can explain without resorting to an outside energy source.
Extended discussion with a simple example
Imagine you have a system which is not in an equilibrium state. How did it get there is irrelevant.
The simplest example is two identical boxes with a perfect gas inside, one at temperature $T_1$, one at temperature $T_2$ with $T_1>T_2$. If they are put somehow in contact, then, without any additional external work, heat will flow from the hot to the cold bath ($T_1\to T_2$) until the two temperatures are the same at the equilibrium temperature $T_e=(T_1+T_2)/2$.
You did not do anything: just put the two gases in contact and they will spontaneously change their temperature to the equilibrium state.
Now, imagine you want the cold gas to stay cold (a fridge) despite its hot sourrounding: to counter the spontaneous process you need to remove heat from the cold gas to keep it cold. To do that, you need some work (the energy given by the current to the fridge).
In this latter case, the entropy of the universe will increase, but the entropy of the sub-system "fridge" will not, because you are artificially keeping it constant. You are basically removing entropy from the fridge at the expense of the universe. But you need to assume that there is an outisde universe that is providing the extra energy required!
Let's reverse the reasoning. You see two gases together, at $T_1$ and $T_2$. You observe them and see that their temperature does not change. Then it must mean that there must be somewhere some work given to the system to maintain them as they are. On the other hand, if you work at the two gases and see that they equilibrate, then you can assume the system is behaving as it should be if there is no work done on it.
That a process is spontaneous means you can describe it without resorting to an external energy source: that is how two isolated gas should behave.
If now you include the whole universe then of course you can explain why the fridge stays cold: the universe already contains the energy source, so there is no inconsistency. And because the universe is isolated, you don't see any process inside the universe that requires an energy source outside your system: it's already somewhere in your system.
Summing up, (not-)"spontaneous" actually can only be applied to sub-systems. If you consider only a subsystem, which can exchange heat with the rest of the universe (so not an isolated system) you can make it evolve towards states which the same subsystem, if isolated, would never evolve to. We call spontaneous processes those that would happen also if the system was completely isolated.
In order to have a non - spontaneous process, you need some sort of energy flux (work, heat..) from the outside. In the big picture of the universe, on the other hand, everything is spontaneous, in a way.
It really depends on your point of view when describing the system.