What is the intuition behind time being a physical dimension? I read the phrase "in cosmology terms, far away means long ago" somewhere and I got to thinking what if time is an emergent property of the interaction between matter and gravity. How can we think of time outside of space and matter? Or does it not need to be? Or am I just confusing matter and physical dimensions?
The intuition behind time being a physical dimension is in the experience of being able to "travel" through it. However, in daily life we only seem to be able to move in one direction and at one speed. In theory, we enter the realm of traveling through time forward, backward, and at different speeds. Thus it has traits of physical dimensions. The phrase, "in cosmology terms, far away means long ago", can at least refer to what we see. Because light takes a specific amount of time to reach us, we are actually seeing an image of something which existed in the past. The farther away it appears to be, the older the image which we see. One way we can think of time outside of space and matter is by thinking of space and matter being inside time. Time does not necessarily need space or matter to exist. If space and matter exist, they do so for a length of time. Matter exists in the physical dimensions. The physical dimensions describe matter. Thus, there are two uses of the term "physical dimensions". The first is the name of a place, the second is a measurement.
There are various theories whereby spacetime (the whole ball of wax) is emergent, e.g., https://arxiv.org/abs/1504.00464 Reading between the lines of your question, I'd guess the best way for you to think about it intuitively is as follows. What's more fundamental: "objects" or "events"? To best interpret time as just another dimension, instantaneous "events" would be the better answer. Then, say, "eating breakfast" and "eating lunch" are two events, separated not only by the distance between the two restaurants you ate at, but by the several hours difference when you ate the meals. In this kind of view, "objects" are just highly-correlated (and kind-of-continuous) sequences of events.