There are a number of different readability tests, depending on what you think is the best measure for your purposes. Most of them are structured around length of sentences, density of text and the use/choice of vocabulary. You probably heard of the Flesch Kincaid readability test used in Microsoft Word if you have used spell check in the old versions.
There are two aspects of readability to consider, the first being the amount of time it takes to read through the material, and the second aspect being the comprehension of that material (which may or may not be related to the amount of time taken to read).
You can measure the first by defining a start and end point for the reading task (e.g. by asking them to do something after they finish reading) and compare the difference in time between the two blocks of text.
And you can measure the second by creating a task that can be completed successfully if they have understood all the materials that they have been asked to read. Once again you would compare between the two blocks of text but not with the same person.
Once you have collected enough data from enough users to make you feel confident enough about the significance of the results, you can summarize and put a figure on the amount of improvement on time and task completion rate to show the impact of the change suggested.