How did Variety know how many families went to see "Avengers: Age of Ultron"?

I feel that it's a large assumption to conclude that Variety got their statistics from a particular company as they do note cite their sources within their article (that I can see). There are many companies that deal with this kind of surveying and while they will all ask a core set of questions (gender, age, ethnic background, etc) some of them will go into more detail than others.

A likely contender for where the particular statistics mentioned in the Variety article is PostTrak, by Rentrak/Screen Engine. Their methodology is the polling of a statistically significant portion of moviegoers - examples of the sort of statistics they collect and their methodology is discussed in this MPAA Theatrical Market Statistics 2014 document. This isn't specific to PostTrak either, this is generally how the system works regardless of whether you're talking about movies or games or laundry detergent.

Nielsen, who collect ratings for TV shows, call this "Audience Composition", but it is essentially demographic analysis. Short of counting and actively monitoring the people entering cinemas (which would be plausible but probably significantly more work than the value these figures themselves provide) all of these numbers are extrapolated.

This article on the Neilsen website: "Popcorn People: Profiles of the U.S. Moviegoer Audience" details the sort of breakdowns by demographic that audience measurement companies look for but also details their process for working out the numbers:

A two-stage study in which online, phone, and in-person surveys were conducted during August and September 2012 among more than 3,000 Americans aged 12-74, who were nationally representative of the U.S. population of moviegoers by age, gender, and race. “Moviegoer” is defined as someone having attended at least one movie in a theater in the past 12 months.

Another example within the industry would be Cinema Score, as mentioned within your question, who state that their process is basically the same as Neilsen's:

For over 35 years, CinemaScore has been polling moviegoers at major movie releases on opening night to collect demographic information and calculate a distinctive CinemaScore grade.

Followed by:

...while a movie critic only provides a single perspective on a movie, a statistically robust sample of a national audience offers a broader and more varied point of view.

Audience members fill out ballot cards right at the theatre, grading a movie A to F and providing demographic information.

What Variety are actually saying is "based on questioning a sample size of X and then extrapolating these numbers up to the estimated seats sold, roughly 59% of the audience was male, 41% was 25 years and older, 12% were teenagers and 22% were families."