Do the relativistic effects of time dilation, length contraction, and mass increase explain the universal measured speed of light or does the universal measured speed of light necessitate those relativistic effects?
The historical answer is definitely that a constant speed of light was noticed in the Maxwell equations, it was taken as a hypothesis that the speed was indeed constant, and the relativistic effects that you mention (among others) were derived. Further experiments have validated the hypothesis by observing the predicted effects.
Special relativity in self-consistent, however, so, from that perspective, you can take the relativistic effects as given and infer the constant speed of light. The historical order that things were observed doesn't change that.