Meaning of 'angle' in angle modulation
Well, the Angle modulation basically means adding a phase function to the carrier that depends on time. 

Let's call that instantaneous phase deviation function as:

$$
\Phi(t)
$$

and our modulated signal is

$$
m(t) = A(t) * cos(\omega t + \Phi(t)).
$$

As you can see, if we eliminate the amplitude modulation (constant A(t)), we have a phase shift that varies with time in the modulated signal formula. And hey, why the heck we don't use this phase shift to transmit our information? It can be noise resistant and easy to demodulate at the same time! It's a terrific idea!

So, if we come to your question, the answer is obvious. We are modulating our information to carrier's instantaneous phase function, therefore it's angle.

If you are looking for further information and modulation techniques, you can take a look at frequency shift keying or phase shift keying modulations. Angle modulation is not limited to these modulation techniques, however they are easy to understand and give you a deeper understanding about angle modulation.

As always, have a nice day.