In terms of physics, does the phrase "time slows down" mean the same thing as "things happen more slowly?" The common definition of "time" is a type of measurement, like size. But the sentence "size gets bigger" doesn't make any sense. Is "time slows down" an odd phrasing of "events occur more slowly" or is there a deeper meaning to the phrase?
 A: 
The common definition of "time" is a type of measurement, like size.

No. The common definition of "time", certainly in the context of physics, is as one indication of one participant, or also as the ordered set of all indications of one participant. As Einstein put it: 
"[... that instead] of "time" we substitute "the position of the little hand of my watch"." [Punctuation marks conforming to the German original: Ann. Phys. 17, 891 (1905)]
The measure of "time" (in the sense of an ordered set of indication of one particular participant, referring to one specific initial indication and one specific final indication) is instead called the "duration" (or also, less distinctively, the "proper time") of that participant, with respect to his/her/its specified initial and final indications.

But the sentence "size gets bigger" doesn't make any sense. 

Well, actually it does, if referring to comparison of values of "size" (i.e. some measures of spatial extension) obtained in different, subsequent trials.

Is "time slows down" an odd phrasing of "events occur more slowly" or is there a deeper meaning to the phrase?

No: the phrase "time slows down" is actually quite pointless itself, and at best an improper way of referring to "decreasing duration, under the condition of equal (proper) rate".
A: In terms of physics time is a coordinate and defines a coordinate system . In the newtonian world we live our lives in, time is fixed by clocks and space by rulers.
Slowing or acceleration of time is a personal perception, old people feel time passes very fast, in crisis situations it passes very slowly in the observer's perceptions.
If one goes into special relativity, both space and time may change according to the system of reference of the observer, but that is another story, affecting elementary particles and cosmology.
There is no meaning in terms of physics to either phrase. In terms of living it is the second sentence that explains the first.
A: It means someone in a different frame of reference would observe events happening slowly in the other frame of reference. 
All observers in their own frame of reference perceive time to be flowing at the same rate. Differences only arise when observing different frames of reference.
The phrasing is an oddity and peculiarity of English. It is imprecise. It only makes sense because it's more commonly heard. Thus it's more colloquial, and seemingly more simplistic to the average ear. It's idiomatic, but it's not precise at all.

Very perceptive question. On an unrelated, but similar note, I always found the phrase "observe" to be very confusing and imprecise. What fundamentally, physically defines "observing"? A human watching? particle entanglement? It turns out, that Feynman, and other documentaries make this clear. "observing" for humans requires light, at least currently. Thus observing fundamentally means we bombard it with a photon, and detect the reflection of that photon with whatever instrument. Now, it makes more sense that the interaction with a photon disturbs the system. And the momentum of the photon is what gives rise to the uncertainty principle and the constant it contains (plank's constant), which shows up in the equation for the energy of light. 
So "observing" doesn't really have to do anything with it, since it's such an abstract, high level, non-fundamental, human concept. Such English phrases like that really obscure the fundamental mechanisms for those new to the concept, and imo is lazy usage of language.
