In this context, the dimensions are the degrees of freedom that a particle can move in. I think it becomes meaningful when you consider the metric (ie. definition of distances) and the laws of nature inside that space, and if additional dimensions help in formulating more phenomena with a less complex system of equations.

-> Classically, there are 3 spatial dimensions (directions of movement) with Euclidean metric, and all the classical laws (Newton's laws, gravity, etc.). Quite successful, but breaking down in extreme cases (high energy, ie. velocity/mass/etc). +there is a whole range of seemingly unrelated laws necessary.

-> with relativity theory (both special and general), time becomes a 4th dimension, but it has a special status in the metric (opposed sign), making spacetime non-Euclidean. One effect of this is suddenly you don't need a law of gravity anymore. Things just follow geodesics in this space. It also explains some effects that could not be derived from classical gravity. Essentially explaining more phenomena with less "overhead", in exchange for more dimensions.